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Geometry - High School

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New York Learning Standards

 

Geometric Relationships

 

Constructions Locus
 

Proofs

Informal & Formal

Transformational Geometry Coordinate Geometry
 

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Geometric Relationships

 

Pairs of Lines Two lines can be related to each other in four different ways. Click on each diagram to learn more about pairs of lines.

 

Undefined Terms and Intuitive Concepts In geometry, definitions are formed using known words or terms to describe a new word.  There are three words in geometry that are not formally defined.  These three undefined terms are point, line and plane. There are a few basic concepts in geometry that need to be understood, but are seldom used  as reasons in a formal proof.

 

Points, Lines, Planes This 15 problem worksheets covers some basics terms and postulates of geometry.  Vocabulary includes point, line, plane, collinear, and coplanar.

Parallel LinesPerpendicular Lines Two lines are perpendicular if the product of their slopes is -1. Also, the two intersecting lines form right angles.

 

Perpendicular Lines Two lines that meet at a right angle are perpendicular.

 

Parallels and Perpendiculars Mathematics problems often deal with parallel and perpendicular lines. Since these are such popular lines, it is important that we remember some information about their slopes.

Perpendicular Planes Two planes are perpendicular if and only if any line in one of them that is perpendicular to their line of intersection is also perpendicular to the other plane.

Parallel and Perpendicular Planes Parallel planes are two planes that do not intersect. A plane B is perpendicular to a plane A if plane B contains a line that is perpendicular to plane A.

Plane Geometry - Perpendiculars Can you solve this problem? Can you draw a perpendicular line from C to AB?

 

Planes: Parallel, Perpendicular and Otherwise Planes have no bumps and like lines go on forever. Three (noncollinear) points determine a plane.

 

Types of Lines When two lines exist within a single plane we say that they are coplanar. Click on Fig.2 to see a plane (in the form of a grid) appear. Clicking and dragging on the grid will rotate the plane about the red line.

Coplanar Geometric objects lying in a common plane are said to be coplanar.

Euclid's Postulates Here are Euclid's 5 postulates, the foundation for plane geometry. 

Euclid's Postulates If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two Right Angles, then the two lines inevitably must intersect each other on that side if extended far enough.

Parallel Postulate Given a line and a point not on the line, how many lines parallel to the first line can be drawn through the point? If you draw the line and the point on a piece of paper, it seems obvious that the answer is one.

Euclid's Fifth Postulate Besides 23 definitions and several implicit assumptions, Euclid derived much of the planar geometry from five postulates.

Parallel Parallel is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. The existence and properties of parallel lines are the basis of Euclid's parallel postulate.

Space figures A polyhedron is a three-dimensional figure that has polygons as its faces. We can relate some polyhedrons--and other space figures as well--to the two-dimensional figures that we're already familiar with. For example, if you move a vertical rectangle horizontally through space, you will create a rectangular or square prism.

Space Figures and Basic Solids A space figure or three-dimensional figure is a figure that has depth in addition to width and height. Everyday objects such as a tennis ball, a box, a bicycle, and a redwood tree are all examples of space figures. Some common simple space figures include cubes, spheres, cylinders, prisms, cones, and pyramids.

multicolor space figure title image Space Figures Space figures are three dimensional figures. Many of them have various numbers of faces, which is a side of the space figure, vertex or vertices (plural), edges which is where two faces of the space figure meet and vertex or vertices (plural) which are the corners of the various space figures.

Space Figures Quiz Pick the correct answer for each question about space figures.

Three-Dimensional Figures In this unit we'll study three types of space figures that are not polyhedrons. These figures have curved surfaces, not flat faces.

Quia - Three Dimensional Figures Vocabulary Play matching and flash cards games on three dimensional figures.

Shape and Space in Geometry -- Shadows Can you judge an object by its shadow? In this activity you will be asked to determine if a shadow can be produced by a particular shape.

Solid Geometry - ShockWaved Create your own three dimensional shapes such as regular polyhedra, prisms and pyramids. From the main menu select the shape you want to explore.

Prism Page A prism has two parallel faces, called bases, that are congruent polygons. Two prisms will have equal volumes if their bases have equal area and their altitudes (heights) are equal."

Prism A prism has at least two congruent (same size and shape) faces that are parallel to one another. These parallel faces are called bases of the prism, and are often associated with its top and bottom.

Prisms and Cylinders A prism is a polyhedron whose faces consist of two congruent polygons lying in parallel planes and a number of parallelograms.

 

Prism and Pyramids The volume of a pyramid is one third the area of the base times the altitude. Write equations for the volumes of the two pyramids to compare. What is the altitude of the pyramid with a lateral side of the prism as its base?

Prism A prism is a space figure with two congruent, parallel bases that are polygons.

3D Shapes - Prisms This site has definitions, surface area, volume, and examples of prisms.

Right Prism A prism which has bases aligned one directly above the other and has lateral faces that are rectangles.

Right Regular Prism Calculator This happens to be a right regular prism - a solid geometric figure whose bases are parallel regular polygons and whose faces are all rectangles perpendicular to the bases.

Learning About Length, Perimeter, Area, and Volume This two-part example illustrates how students can learn about the length, perimeter, area, and volume of similar objects using dynamic figures.

Volume of a Prism The volume of a prism is a product of the area of the base and the height of the prism.

Volume of a Rectangular Prism The volume of a rectangular prism can be found by the formula: volume = length * width * height.

 

Volume of a Triangular Prism The volume of a triangular prism can be found by the formula: volume = 1/2 * length * width * height.

Pyramid A pyramid is a space figure with a square base and 4 triangle-shaped sides.

3D Shapes - Pyramids This site has definitions, surface area, volume, and examples of pyramids.

Pyramids A pyramid is a polyhedron with a single base and lateral faces that are all triangular.  All lateral edges of a pyramid meet at a single point, or vertex.

Three Dimensional Figures: Pyramids and Cones Pyramids have lateral edges which connect vertices of the base polygon with the vertex. In a cone, the lateral edge is any segment whose endpoints are the vertex and a point on the base circle. The triangular, non-base, faces of a pyramid are lateral faces.

Volume of a Pyramid A pyramid has a base and triangular sides which rise to meet at the same point. The base may be any polygon such as a square, rectangle, triangle, etc.

Volume of a Pyramid Enter the base and height of the pyramid and click the Math button to calculate the volume.

Cylinder A cylinder is a space figure having two congruent circular bases that are parallel. If L is the length of a cylinder, and r is the radius of one of the bases of a cylinder, then the volume of the cylinder is L  pi  r2, and the surface area is 2  r  pi  L + 2  pi  r2.

Cylinders, Cones, and Spheres In this lesson, we study some common space figures that are not polyhedra. These figures have some things in common with polyhedra, but they all have some curved surfaces, while the surfaces of a polyhedron are always flat.

 

3D Shapes - Cylinders This site has definitions, surface area, volume, and examples of cylinders.

 

Cone A cone is a space figure having a circular base and a single vertex. If r is the radius of the circular base, and h is the height of the cone, then the volume of the cone is 1/3  pi x r2  h.

3D Shapes - Cones This site has definitions, surface area, volume, and examples of cones.

The Geometry of the Sphere On the sphere we have points, but there are no straight lines --- at least not in the usual sense. However, straight lines in the plane are characterized by the fact that they are the shortest paths between points.

Sphere A sphere is a space figure having all of its points the same distance from its center. The distance from the center to the surface of the sphere is called its radius.

 

 

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Constructions

Compass and Straight Edge In his Elements, Euclid laid the foundations of mathematics based solely on physical tools, straightedge and drawing compass. This site offers virtual straightedge and compass, through a Java applet named after Euclid. Using virtual straightedge and compass our Euclid applet can draw lines and circles.

How to Use a Geometric Compass A compass is a handy drawing tool to have around. Use it to draw circles, make equal line segments or find the midpoint of a line.

Basic Geometric Constructions In geometry, constructions utilize only two tools - the straightedge (an unmarked ruler) and the compass. Never draw freehand when doing a construction!

 

{short description of image} Geometry Construction Course The aim of these resources is simple: to enable you to develop your skills of geometrical construction. You will be able build mathematical shapes out of points, lines and circles. Then you can change the diagrams yourself using the mouse and see the effect these changes have on the mathematical properties of the shapes.

Interactive Drawing Tool Use this interactive tool to create dynamic drawings on isometric dot paper. Draw figures using edges, faces, or cubes. You can shift, rotate, color, decompose, and view in 2‑D or 3‑D.

Quia - Tools of Geometry This activity covers bisectors, line segments, parallel lines, angles, planes, and more.

Basic Constructions This site covers constructing the perpendicular bisector of a segment, constructing the bisector of an angle, and copying an angle.

 

 Bisecting Segments and Angles Students will learn to identify bisectors of segments and angles in order to find distances and angles measures.

Bisecting Angles and Line Segments The following images provide steps for bisecting angles and bisecting line segments.

Quia - Bisecting Angles and Segments This activity covers bisecting angles and segments.

Copying a Line Segment and an Angle Use a compass and straight edge to construct a line segment congruent to line segment AB, and to construct an angle congruent to angle BAC.

Bisecting a Segment This site has step-by-step instructions on creating a perpendicular bisector of a line segment.

 

compass To Bisect a given Line Segment On the sketch below drag point A around; do you always have a bisector? Does it matter whether A is to the right or left of B?

 

Constructing Perpendicular Lines Given the line through the points A and B, and a point C not on the line, carry out the following steps to construct a line through C perpendicular to Line AB.

 

Constructing Perpendicular Lines This site shows how to construct a line perpendicular to a given line from a point not on a line and a point on a line.

 

Perpendicular Bisector This site has step-by-step instructions on constructing a perpendicular bisector.

 

Perpendicular Bisector Construction This is how you split a segment into two congruent parts. You need a segment to start with.

 

Parallel and Perpendicular Lines Mathematics problems often deal with parallel and perpendicular lines. Since these are such popular lines, it is important that we remember some information about their slopes.

Parallel Lines On this page, we hope to clear up problems that you might have with parallel lines and their uses in geometry.  Parallel lines seem rather innocent, but are used in some complex geometry situations to help you solve problems.  Click any of the links to start understanding parallel lines better! The site includes explanations, tutorials and quizzes.

Constructing Parallel Lines Given the line through the points A and B, and a point C not on the line, carry out the following steps to construct a line through C parallel to Line AB.

 

Proving Lines Parallel Using our prior constructions dealing with angles we are going to construct parallel lines given a line and a point not located on the line.

Constructing an Equilateral Triangle Given the line segment extending from point A to point B, carry out the following steps to construct an equilateral triangle with base AB.

Construct an Equilateral Triangle Construct a third point C, such that the triangle ABC is equilateral.

 

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Locus

Points, Lines, and Circles Associated with a Triangle There are hundreds of different constructions that find a special point inside a triangle, satisfying some unique property.

Median (Triangle) The cevian from a triangle's vertex to the midpoint of the opposite side is called a median of the triangle.

 Median of a Triangle A median of a triangle is a line from a vertex of the triangle to the midpoint of the side opposite that vertex. Drag the orange dots on each vertex to reshape the triangle. Notice the three medians all meet at one point.

Altitude of a Triangle The altitude of a triangle is a line segment from one vertex of a triangle to the opposite side so that the line segment is perpendicular to the side.

All About Altitudes In a triangle, an altitude is a segment of the line through a vertex perpendicular to the opposite side. An altitude is the portion of the line between the vertex and the foot of the perpendicular.

Bisector of an Angle of a Triangle Construct any triangle. Construct an angle bisector in the triangle and draw the segment along the angle bisector from the vertex to the intersection with the opposite side.

Angle Bisectors of a Triangle Prove that the three angle bisectors of the internal angles of a triangle are concurrent.

Perpendicular Bisector of a Side of a Triangle The definition of the perpendicular bisector of a side of a triangle is a line segment that is both perpendicular to a side of a triangle and passes through its midpoint.

The Concurrency of the Three Perpendicular Bisectors of the Sides of a Triangle When three or more lines intersect in the same point, they are called concurrent lines. The point of intersection of the lines is called the point of concurrency. Let's examine the point of concurrency for the perpendicular bisectors of various triangles.

Locus At a Fixed Distance from a Point A locus is a set of points which satisfies a certain condition. Think of a locus as a "bunch" of points that all do the same thing.

Locus at a Fixed Distance from a Line The locus of points at a fixed distance, d, from a line, l, is a pair of parallel lines d distance from l and on either side of l.

Locus Equidistant from Two Points The locus of points equidistant from two points, P and Q, is the perpendicular bisector of the line segment determined by the two points.
 

Locus Equidistant from Two Parallel Lines The locus of points equidistant from two parallel lines, l1 and l2 , is a line parallel to both l1 and l2 and midway between them.

Locus Equidistant from Two Intersecting Lines The locus of points equidistant from two intersecting lines, l1 and l2, is a pair of bisectors that bisect the angles formed by l1 and l2.

Compound Locus If TWO conditions exist in a problem (a compound locus), complete steps 2-4 listed below for EACH of the conditions independently ON THE SAME DIAGRAM.

Circles and Loci When you want to represent a circle on a coordinate plane, you need to use the following equation: (x-h)^2 + (y-k)^2 = r^2, where h and k are the center points of the circle and r is the radius or the circle.

 

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Informal & Formal Proofs

SparkNotes Geometric Proofs: The Structure of a Proof  Geometric proofs can be written in one of two ways: two columns, or a paragraph. A paragraph proof is only a two-column proof written in sentences. However, since it is easier to leave steps out when writing a paragraph proof, we'll learn the two-column method.

Writing Proofs The first step towards writing a proof of a statement is trying to convince yourself that the statement is true using a picture. This will give you a feeling for the statement but is not a proof in and of itself.

Theorem/Properties Sheet for Proofs This is a partial listing of the more popular theorems, postulates and properties
needed when working with Euclidean proofs.  You need to have a thorough understanding of these items.

Vocabulary Resource Sheet for Proofs This is a partial listing of basic FORMAL definitions needed when working with Euclidean geometry and proofs. You need to have a thorough understanding of these terms.

Euclidean Direct Proofs A proof is a written account of the complete thought process that is used to reach a conclusion.  Each step of the process is supported by a theorem, postulate or definition verifying why the step is possible. In formal Euclidean proofs,  no steps can be left out.

Coordinate Geometry Proofs Coordinate geometry proofs employ the use of formulas such as the Distance Formula, the Slope Formula and/or the Midpoint Formula as well as postulates, theorems and definitions.

Coordinate Geometry Proofs Use coordinate geometry to prove that the midpoint of the hypotenuse of a right triangle is equidistant from the vertices of the triangle.

Quia - Geometry Proofs Vocabulary This site has a list of terms used in geometric proofs and explanations of each.

Indirect Euclidean Proofs When trying to prove a statement is true, it may be beneficial to ask yourself, "What if this statement was not true?" and examine what happens.  This is the premise of the Indirect Proof or Proof by Contradiction.

Negation, Conjunction, & Disjunction Watch this slideshow on negation, conjunction, and disjunction.

Conditional Proof A conditional proof is a proof that takes the form of asserting a conditional, and proving that the premise or antecedent of the conditional necessarily leads to the conclusion.

Conditional Proofs Given an argument with a conditional conclusion, assume the antecedent of the conclusion and derive the consequent:

Conditional Statements This worksheet contains introductory questions on conditional statements and converses.  Students are asked to identify the parts of a conditional statement (hypothesis & conclusion) and to find converses of conditional statements.

Biconditional A biconditional is a truth function that is true only in the case that both parameters are true or both are false.

Logical Biconditional In mathematics, logical biconditional is a logical operator connecting two statements to assert, p if and only if q where p is a hypothesis (or antecedent) and q is a conclusion (or consequent).

Biconditional Statements This 14 problem worksheet covers biconditional statements, good definitions, and counterexamples. 

Reasoning Properties This worksheets begins with a review of properties from algebra (reflexive, symmetric, transitive, etc) and wraps up with a couple basic proofs using these properties.

Logic - Related Conditionals People will often "twist words around" in an attempt to win an argument.  In order to avoid becoming a victim of such tactics, you need to be aware of three new conditionals and their related truth values.

Logic - Converse The converse of a conditional statement is formed by interchanging the hypothesis and conclusion of the original statement.

Logic - Inverse The inverse of a conditional statement is formed by negating the hypothesis and negating the conclusion of the original statement.

Logic - Contrapositive The contrapositive of a conditional statement is formed by negating both the hypothesis and the conclusion, and then interchanging the resulting negations.

Practice with Logic and Related Conditionals Answer the following questions dealing with logic and related conditionals.

Tim's Triangular Page A triangle is a closed plane figure bounded by three straight lines meeting at three different points. The three intersection points are triangle vertices. The line segments between the vertices are triangle sides. A triangle can also be defined as a three-sided polygon.

Vocabulary Practice - Basic Triangle Proofs Match each term on the left with its best description on the right. There is only one appropriate answer among the choices on the right. Make your selection for each match to the left of the statement.

Congruence You walk into your favorite mall and see dozens of copies of your favorite CD on sale.  All of the CDs are exactly the same size and shape. In fact, you can probably think of many objects that are mass produced to be exactly the same size and shape.

Congruent Triangles On this page, we hope to clear up problems that you might have with proving triangles congruent.  Triangles are one of the most used figures in geometry and beyond (engineering), so they are rather important to understand.  Scroll down or click any of the links to start understanding congruent triangles better! The site includes explanations, tutorials and quizzes.

Congruent Right Triangles On this page, we hope to clear up problems that you might have with proving right triangles congruent.  Right triangles are special triangles that contain one right angle.  With right triangles, we name the sides of the triangle.  The two sides that include the right angle are called legs and the side opposite the right angle is called the hypotenuse.  Scroll down or click any of the links to start understanding congruent right triangles better! The site includes explanations, tutorials and quizzes.

Interior Angles of a Triangle The sum of the interior angles of any triangle is 180.

The Sum of Three Interior Angles Why is the sum of three interior angles 180 degrees? People just use this property in many geometry problems without considering this fact. This seems very basic and simple but is very powerful tool in problem solving.

Isosceles Triangle Theorem In the figure, ABD is an isosceles triangle with AB = BD. Also AE = CD. F is the intersection of AD and EC. The theorem states that EF = FC.

Isosceles Triangle Theorem The isosceles triangle theorem states that if two sides of a a triangle are congruent then the angles opposite of the sides are congruent. The converse of this statement is also true.

ACD is an exterior angle of triangle ABCExterior Angle of a Triangle An exterior angle of a triangle is formed when any side is extended outwards.

The Exterior Angle Theorem The Exterior Angle Theorem, "In any triangle, if one of the sides is produced, then the exterior angle is greater than either of the interior and opposite angles,"  is one of the cornerstones of elementary geometry.

Exterior Angle Theorem An exterior angle of a triangle equals the sum of the two interior opposite angles in measure.

Triangle Inequalities The sum of the lengths of any two sides of a triangle must be greater than the third side.

Triangle Inequality Theorem The Triangle Inequality Theorem says. . .The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

The Triangle Inequality Theorem Applet This applet demonstrates that if the process of taking three random length line segments and attempting to form a triangle with those segments is repeated many times, while keeping a tally of each outcome, the relative frequency of the successes will eventually become very close to .25.

Proportionality of Triangles In every triangle, the longest side is opposite the largest angle and the shortest side is opposite the smallest angle.

Proving Lines are Parallel If two parallel lines are cut by a transversal, then the corresponding angles are congruent. If two lines cut by a transversal have congruent corresponding angles, then the lines are parallel.

Parallel Line Property The Parallel Lines Property states that if two lines are cut by a transversal such that alternate interior or exterior angles have equal measure, then the two lines are parallel. It also states that two lines cut by a transversal are parallel if and only if corresponding angles have the same measure.

Interior Angles of Polygons The sum of the measures of the interior angles of a triangle is 180 degrees. Since all the sides of equilateral triangles are the same length, all the angles are the same...

Sum of Interior Angles of a Polygon The formula we use to find the sum of the interior angles of any polygon comes from the following idea. . .

Interior Angles of Polygons An Interior Angle is an angle inside a shape. The Interior Angles of a Triangle add up to 180.

Exterior Angles An exterior angle of a polygon is formed by extending one side of the polygon.

Parallelogram A parallelogram is a four-sided polygon with two pairs of parallel sides. The sum of the angles of a parallelogram is 360 degrees.

Parallelogram A parallelogram is a quadrilateral that has two pairs of parallel sides. The opposite sides are equal in length.

parallelogram The Properties of a Parallelogram Definitions and formulas for the perimeter of a parallelogram, the area of a parallelogram, properties of the sides and angles of a parallelogram.

Exploring Properties of Rectangles and Parallelograms Dynamic geometry software provides an environment in which students can explore geometric relationships and make and test conjectures. In this example, properties of rectangles and parallelograms are examined.

rectangle The Properties of a Rectangle Definitions and formulas for the perimeter of a rectangle, the area of a rectangle,  how to find the length of the diagonal of a rectangle, properties of the diagonals of a rectangle.

Rhombus A rhombus is a four-sided polygon having all four sides of equal length. The sum of the angles of a rhombus is 360 degrees.

rhombusThe Properties of a Rhombus Definitions and formulas for the perimeter of a rhombus, the area of a rhombus,  properties of the angles and sides of a rhombus.

Definition of a Square The square is probably the best known of the quadrilaterals. It is defined as having all sides equal, and its interior angles all right angles (90). From this it follows that the opposite sides are also parallel.

squareThe Properties of a Square Definitions and formulas for the perimeter of a square, the area of a square,  how to find the length of the diagonal of a square, properties of the diagonals of a square.

The TrapezoidThe Trapezoid The trapezoid is a quadrilateral with one pair of parallel sides.

trapezoid Trapezoids Definitions and formulas for the perimeter of a trapezoid, the area of a trapezoid, properties of the sides and angles of a trapezoid, properties of the sides and angles of an isosceles trapezoid.

The Quadrilateral Family Meet the quadrilateral family. I have exactly four sides.

Quadrilaterals On this page, we hope to clear up problems that you might have with quadrilaterals.  Quadrilaterals are the most used shape (they cover everything from squares to trapezoids) in geometry except for the triangle.  Scroll down to start understanding quadrilaterals better! The site includes explanations, tutorials and quizzes.

Quia - Quadrilateral Properties Match the quadrilaterals with their defining properties.

The Midpoint Theorem The segment joining the midpoints of two sides of a triangle is parallel to the third side and half as long as the third side.

Centroid of a Triangle The centroid of a triangle is the point of intersection of the three medians. A median is the segment from a vertex of the triangle to the midpoint of the opposite side.

a triangle, showing its medians Finding the Centroid of a Triangle Prove, using a vector method, that the medians of a triangle meet at a point which divides each median in the same ratio. Find this ratio. This point is called the centroid of the triangle.

Similar Triangles In mathematics, polygons are similar if their corresponding (matching) angles are equal and the ratio of their corresponding sides are in proportion.

Similar Triangles If two shapes are similar, one is an enlargement of the other. This means that the two shapes will have the same angles and their sides will be in the same proportion (e.g. the sides of one triangle will all be 3 times the sides of the other etc.).

Similar Triangles Two triangles are similar if their corresponding angles are congruent and their corresponding sides are proportional.

Accompanying FigureParallel Lines and Triangles If a segment is parallel to one side of a triangle and intersects the other sides in two points, then the triangle formed is similar to the original triangle.  Also, when you put a parallel line in a triangle, as the theorem above describes, the sides are divided proportionally.

Side Splitter Theorem PictureSide Splitter Theorem - Similar Triangles and Parallel Lines The side splitter theorem states that if a line is parallel to a side of a triangle and intersect the other two sides, then this line divides those two sides proportionally.

Right Triangles Let's agree again to the standard convention for labeling the parts of a right triangle. Let the right angle be labeled C and the hypotenuse c. Let A and B denote the other two angles, and a and b the sides opposite them, respectively.

Right Triangles The right triangle has been used in trades for thousands of years. Ancient Egyptians found that they could always get a square corner using the 3-4-5 right triangle.

Right Triangle Trigonometry Certain triangles possess "special" properties that allow us to use "short cut formulas" in arriving at information about their measures.  These formulas let us arrive at the answer very quickly.

Quia - Triangle Review Quiz Triangle classification, terminology, congruence postulates/theorems.

Right Triangle Facts The right triangle is one of the most important geometrical figures, used in many applications for thousands of years.

The Pythagorean Theorem The Pythagorean Theorem was one of the earliest theorems known to ancient civilizations. The Pythagorean Theorem is a statement about triangles containing a right angle. The Pythagorean Theorem states that: "The area of the square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides."

NOVA Online NOVA Online: Pythagorean Theorem Puzzle On the diagram, show that a2 + b2 = c2, by moving the two small squares to cover the area of the big square.

Pythagorean Theorem Annotated Animated Proof of the Pythagorean Theorem Check out this animated site that proves the Pythagorean Theorem.

Circle Geometry Proofs This page offers formal proofs for the derivation of circles tangential to other graphic elements.

Ask Dr. Math - Circle Formulas Learn about arcs, segments, sectors and more.

Circle Geometry You will be using three Java Applets created by Ron Blond to investigate various aspects of circle geometry. You will use the applets to develop and apply the geometric properties of circles and polygons to solve problems.

Circle Geometry: Gap-Fill Exercise Fill in all the gaps, then press "Check" to check your answers. Use the "Hint" button to get a free letter if an answer is giving you trouble.

Quia - Circles Word Games Test your knowledge of circle terms playing Matching, Concentration, and Word Search.

Perpendicular Bisector of a Chord The chord of a circle is a segment whose endpoints are on the circle. This conjecture states that the perpendicular bisector of any chord passes through the center of the circle. This could give us a good way to find the center of any circle!

Perpendicular Bisector This site has step-by-step instructions on constructing a perpendicular bisector.

Segment Rules in Circles Chords, Secants, Tangents If two chords intersect in a circle, the product of the lengths of the segments of one chord equal the product of the segments of the other.

Chords_and_Distances The only way to get chord DE congruent to chord LK is to have distance FC equal to distance MC.

Circle Geometry (Tangents and Radii) Drag any highlighted location on the circle ( ) to move that location along the circle. Drag the center of the circle ( ) to re-position the circle.

Line Tangent to a Circle at a Given Point A line that is tangent to a circle at a given point is perpendicular to the radius that intersects the point.

Circles: Tangent Lines and Secant Lines A tangent line is a line that intersects a circle at one point. Such a line is said to be tangent to that circle. The point at which the circle and the line intersect is the point of tangency.

Arc of a Circle Arcs are measured in two ways: as the measure of the central angle, or as the length of the arc itself.

Conjectures in Geometry: Arc Length The arc length of a circle is the distance from one point on the circumference to another point on the circumference, "traveling" along the edge of the circle.

Central Angles and Arcs Central angles are angles formed by any two radii in a circle. The vertex is the center of the circle. An arc of a circle is a continuous portion of the circle. It consists of two endpoints and all the points on the circle between these endpoints.

Circle Geometry (Inscribed and Central Angles) Drag any highlighted location on the circle to move that location along the circle. Drag any non-highlighted location on the circle to change the radius of the circle.

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Transformational Geometry

 

Introduction to Isometries Whenever you move something like a teapot from the cupboard to the table, you have just unwittingly performed a mathematical operation called an isometry. In this 5 minute seminar series, you can find out more about isometries, their connection to geometry, and their application to computer animation.

Isometries The only isometries of the plane are combinations of translations, rotations, and reflections. More specifically, if two figures are congruent, you can transform one into the other first by reflecting (if necessary), then rotating (if necessary), finally translating (if necessary).

Trasformations Review Review reflection, rotation and translation. Then answer questions about what you have learned.

Transformations A transformation is a change in position, shape, or size of a figure. A preimage is a figure to be transformed (the original picture). An image is a figure that has been transformed. An isometry is a transformation in which the preimage and image are congruent.

transformations Interactive Transformations You will be able change various shapes using rotations, reflections, translations or enlargements - and combinations of these transformations. Using your mouse you will drag point and lines or even create new points and lines. Find what mathematical effects these changes have on the symmetries and other properties of the shapes.

Alphabet Geometry: Flips, Slide, Turn The word transform means "to change." In geometry, a transformation changes the position of a shape on a coordinate plane. What that really means is that a shape is moving from one place to another.

Putting Transformations Together In this project you will use the motion of sailboats to explore transformations.

oriental frieze pattern Frieze Frames In this project, you will use geometry to identify frieze patterns that were created using reflections, translations, rotations, and glide reflections.

Transformations and Patterns You will use Islamic art from Spain to further explore frieze patterns.

Using a Graphing Calculator to Examine Transformations This site gives step-by-step instructions on examining transformations using a graphing calculator.

Transformations in Coordinate Geometry - Translations A translation "slides" an object a fixed distance in a given direction.  The original object and its translation have the same shape and size, and they face in the same direction.

Translations A translation is a transformation that moves points the same distance and direction.

  Practice Recognizing Translations Do the diagrams illustrate translations?

Working with Translations Answer the following questions relating to translations. 

Transformations in Coordinate Geometry - Dilations A dilation is a transformation that produces an image that is the same shape as the original, but is a different size. The description of a dilation includes the scale factor and the center of the dilation.

Working with Dilations Answer the following questions dealing with dilations and coordinate geometry.

Basic Transformations - Dilations A dilation is a transformation that extends all points in a figure a given scale through a point.  The resultant figure is either an enlargement of the image (scale > 1) or a reduction of the image (scale < 1).

Transformations: Rotation A rotation is a transformation that turns a figure about a fixed point called the center of rotation.  An object and its rotation are the same shape and size, but the figures may be turned in different directions.

Rotation When a figure is turned, we call it a rotation of the figure. We can measure this rotation in terms of degrees; a 360 degree turn rotates a figure around once back to its original position.

Rotation A rotation is an isometry. A rotation does not change orientation. Rotations can move clockwise (negative) or counterclockwise (positive).

 

Transformations in Coordinate Geometry - Reflection in a Line When you reflect a point across the x-axis, the x-coordinate remains the same, but the y-coordinate is transformed into its opposite.

Reflection If we flip (or mirror) along some line, we say the figure is a reflection along that line.

  Practice Recognizing Reflections Do the diagrams illustrate line reflections?

Working with Reflections Answer the following questions relating to reflections. Check your answers in the answer boxes.

 

 

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Coordinate Geometry

Coordinates and Similar figures Learn all about coordinates in graphing and figures that are similar in shape and size.

Coordinate Geometry On this page, we hope to clear up any problems that you might have with coordinate geometry.  Scroll down or click any of the links below to start understanding coordinate geometry better! The site includes explanations, tutorials and quizzes.

Practice with Area and Coordinate Geometry Answer the following questions dealing with area in coordinate geometry.

Coordinate Geometry In algebra, you were introduced to the coordinate system, plotting ordered pairs, and graphing lines. These tools are used in geometry as well. Algebra and geometry are used hand-in-hand to solve many real-world math problems

Coordinate Geometry This site covers coordinate geometry. Each section begins with a short lesson and then an exercise. Once students have attained a score of 10 they are ready to move on to the next stage.

Equation of a Line When working with straight lines, there are many ways to arrive at an equation which represents the line.

Equations of Lines In this tutorial we will look closer at equations of straight lines. We will be going over how to come up with our own equations given certain information.  We will be using the slope of the line and a point it passes through to do this.

Straight-Line Equations Straight-line equations, or "linear" equations, graph as straight lines, and have simple variables with no exponents on them.

Slope The slope is measured by comparing how much the rise in the road for example over the distance it takes for the rise.

Slope of a Straight Line One of the most important properties of straight lines is their angle from horizontal. This concept is called "slope".

Slope Two lines are parallel if they have the same slope. Two lines are perpendicular if their slopes are opposite reciprocals of each other.

Slope of Perpendicular Lines Perpendicular lines have negative reciprocal slopes.

Parallel LinesPerpendicular Lines Two lines are perpendicular if the product of their slopes is -1. Also, the two intersecting lines form right angles.

 

Perpendicular Lines Two lines that meet at a right angle are perpendicular.

 

Parallels and Perpendiculars Mathematics problems often deal with parallel and perpendicular lines. Since these are such popular lines, it is important that we remember some information about their slopes.

Parallel and Perpendicular lines This tutorial looks at the relationship between the slopes of parallel lines as well as perpendicular lines. You will need to know how to find the slope of a line given an equation and how to write the equation of a line.

Ask Dr. Math - Coordinate Geometry Given two known coordinate points in cartesian plane pt1, given by (x1, y1), and pt2, given by (x2, y2), how can I locate a third point pt3, given by (x3, y3), which is perpendicular to the line joining pt1 and pt2 and is of certain distance d from either pt1 or pt2?
 

Midpoint Formula The point halfway between the endpoints of a line segment is called the midpoint.
A midpoint divides a line segment into two equal parts.

The Midpoint Formula Sometimes you need to find the point that is exactly between two other points. For instance, you might need to find a line that bisects (divides into equal halves) a given line segment. This middle point is called the "midpoint".

Verify that a Midpoint Exists This problem will help to reinforce the idea that there is more than one way to solve a problem. For the graph at the left, find at least two ways to verify that M is the midpoint of segment AC.

Length of a Line Segment The easiest way to show you how to do this is by working out an example. Find the distance between (-2,8) and (-7,-5).

Length of a Line Segment  When working with Coordinate Geometry, there are many ways to find distances (lengths) of line segments on graph paper. Let's examine some of the possibilities.

Ask Dr. Math - Coordinates of a Point ABC is a right-angled triangle labeled counter-clockwise with its point C lying on the line y=3x. A is (2,1) and B is (5,5). Find the two possible coordinates of C.

Ask Dr. Math - Coordinates of a Right Triangle Find all possible values of k so that (-1,2), (-10,5), and (-4,k) are the vertices of a right triangle.

The Rectangular Coordinate System This section covers the basic ideas of graphing:  rectangular coordinate system, ordered pairs and solutions to equations in two variables. 

Ask Dr. Math - Polar Coordinate System vs. Rectangular Coordinate System What is the polar coordinate system and how does it differ from the rectangular coordinate system?

Graphing Systems of Equations If you can graph a straight line, you can solve systems of equations graphically! The process is very easy.  Simply graph the two lines and look for the point where they intersect (cross).

Graphing Systems of Equations A set of equations, for example, two equations with two unknowns, for which a common solution is sought is called a system of equations. When we graph a system of two linear equations, one of three things may happen.

Equations of Circles When the equation of a circle appears in "standard form", it is often beneficial to convert the equation to "center-radius" form to easily read the center coordinates and the radius.

Equation of a Circle In this tutorial we get to look at circles.  We will discuss how to write an equation in standard form given either the radius and center or the equation written in general form.

[Circle Image]Deriving the Standard Equation of a Circle How would we write the equation of a circle whose centre is a point other than the origin? We will follow the same procedure as before to come up with the equation for a circle with radius r whose centre is now any point (h, k) on the coordinate plane.

Equations of Circle - Quiz Answer these online questions about the equation of a circle.

Graphing Circles Write the equation of a circle with center at (3,-5) and a radius of 3.

Graphing Circles Find the equation of the circle centered at the origin, with a radius of 4.

Graphing Circles In this section we are going to take a quick look at circles.  However, before we do that we need to give a quick formula that hopefully youll recall seeing at some point in the past.

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Lesson Plans/Classroom Activities

 

Prisms: Area & Volume

Doppler Radar Activity

Emergency Broadcasting Stations

Locating Locus Using Manipulatives

Using Sequencing with Euclidean Proof

Indirect Proof Warm-Up Activity

Applying Locus Theorem 5

Congruent Triangle Activity

Right Triangle Trigonometry

Surface Area Activity

Coordinate Graphing

Graphing Game

Proving Pi

Discovering Growth Patterns

An Introduction to Angles

Travel Triangles

Finding the Sum of the Exterior and Interior Angles of a Polygon

Polygons Made to Order

Properties of Quadrilaterals

How Many Regular Polyhedrons Are There In This or Any Universe?

Similarity

Circumferences, Diameters, and Radii

The Area of a Circle

Measurement of Volume

The Surface Area of a Cylinder

 

 

L to L: Additional Lesson Plans/Classroom Activities

 

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