Math Forum - Problems Library - Geometry - Similarity, Congruence

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Geometry
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  perimeter, area,
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  similar/congruent
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Similarity and Congruence
These problems require knowledge of the relations between similar or congruent shapes. Some of the solutions will require a proof of the congruency. Others will examine the statement of proportionality between two similar shapes. Many of the problems in this category will require advanced knowledge and an integration of several geometric topics.

Related Resources
Interactive resources from our Math Tools project:
   Geometry: Congruence and Similarity
The closest match in our Ask Dr. Math archives:
   High School: Triangles
NCTM Standards:
   Geometry Standard for Grades 9-12


Access to these problems requires a Membership.

Arda and Brooke's Catapult - Annie Fetter
Geometry, difficulty level 2. Does Arda's method for measuring the distance across the river really work? ... more>>

Area of a Rectangle - Annie Fetter
Geometry, difficulty level 2. Given rectangle ABCD with point E on CD. AE is 3, BE is 4, and AE is perpendicular to BE. What's the area of ABCD? ... more>>

Arranging Rectangles - Annie Fetter
Geometry, difficulty level 3. Help Kasim decide if his arrangement of rectangles could ever form a square. ... more>>

Beach Boys - Steve Risberg
Geometry, difficulty level 3. Help Skip and Gil find the height of the cruise ship. ... more>>

Building a Vaulted Ceiling - Annie Fetter
Geometry, difficulty level 4. An eyebrow window is a 15-inch-deep slice off an 8.5-foot circle. How wide will the base of the window be? ... more>>

A Carpenter's Trisection - Annie Fetter
Geometry, difficulty level 3. Explain whether or not the given method of trisecting an angle, using only a carpenter's square, really works. ... more>>

A Circle Inscribed in an Isosceles Triangle - Ilmar Vitsut
Geometry, difficulty level 4. Find the radius of a circle inscribed in an isosceles triangle with sides 12, 12, and 8. ... more>>

Circles and Tangents - Annie Fetter
Geometry, difficulty level 3. AOD is a diameter of circle O. B is any point on the circle. At B, a tangent is drawn to the circle. From the center, O, a line is drawn parallel to AB, meeting the tangent at P. Prove that PD is tangent to the circle. ... more>>

Clever Cuts - Annie Fetter
Geometry, difficulty level 2. Find three different ways to cut a hexagonal cake into six congruent pieces. ... more>>

Congruence and Area - Annie Fetter
Geometry, difficulty level 3. "If two triangles have the same area, then they are congruent." Is this a true statement? ... more>>

Congruent Chords and Circles - Annie Fetter
Geometry, difficulty level 4. Prove that two congruent chords in a circle are equal distances from the center of the circle. ... more>>

Congruent Rectangles - Annie Fetter
Geometry, difficulty level 1. Seven congruent rectangles are arranged to form a larger rectangle. If the area of the large rectangle is 756 units^2, what's the perimeter of the large rectangle? ... more>>

Congruent Rectangles Help - Annie Fetter
Geometry, difficulty level 2. Find some possible areas for the large rectangle shown in the given picture. ... more>>

Constructing the Great Pyramid - Annie Fetter & Suzanne Alejandre
Geometry, difficulty level 2. Figure out if the instructions provided will result in a scale model of the Great Pyramid of Giza. ... more>>

The Diagonal Triangle - Annie Fetter
Geometry, difficulty level 1. A triangle is formed by connecting opposite vertices of a figure formed by two squares. Find the area of the triangle. ... more>>

Dilating a Quadrilateral - Annie Fetter
Geometry, difficulty level 3. Given two similar quadrilaterals on a coordinate grid, determine the ratio and center of dilation. ... more>>

Drawing a Kite Plan - Annie Fetter
Geometry, difficulty level 3. Given a picture of a kite plan, figure out the radius of the circle that creates the curved section. ... more>>

Equal Area & Congruence - Annie Fetter
Geometry, difficulty level 2. If two triangles are congruent, they have the same area. Is the converse true? ... more>>

Equilateral Triangle and Square Ratio - Dale Pearson
Geometry, difficulty level 3. A square with side length s is inscribed in an equilateral triangle of side length t. Find the ratio t/s. ... more>>

Explanation Only! - Annie Fetter
Geometry, difficulty level 4. Prove that x/(1-x) = 1/x in the given picture. ... more>>

Finding CE - Annie Fetter
Geometry, difficulty level 4. Side AB of triangle ABC has a length of 8 inches. Line DEF is drawn parallel to AB so that D is on segment AC and E is on segment BC. Line AE extended bisects angle FEC. If DE has a length of 5 inches, what's the length of CE? ... more>>

Finding Fireworks - Annie Fetter
Geometry, difficulty level 2. Do Nasheed and Graciella have enough information to figure out where the fireworks show is? ... more>>

Finding the Side of a Parallelogram - Annie Fetter
Geometry, difficulty level 3. In parallelogram ABCD, the bisector of angle ABC intersects AD at point P. PD is 5, BP is 6, and CP is 6. What's AB? ... more>>

Folding a Piece of Paper in Thirds - Annie Fetter
Geometry, difficulty level 3. Show that the given folds mark a piece of paper exactly in thirds. ... more>>

Geometry X-Games - Annie Fetter
Geometry, difficulty level 2. Given triangle ABC, with D on AB and E on BC. AC is 39 units, AD is x units, DB is 9 units, DE is 13 units, and BE is x units. If triangles ABC and EBD are similar, what's x, and what's the length of CE? ... more>>

How Long is the Base? - Annie Fetter
Geometry, difficulty level 3. Find the length of the base of a particular isosceles triangle. ... more>>

An Inscribed Circle - Annie Fetter
Geometry, difficulty level 2. A circle is inscribed in a triangle with sides of 6, 6, and 4. What's the radius of the circle? ... more>>

Kyle's Similar Triangles - Kyle Binder and Annie Fetter
Geometry, difficulty level 3. Given two similar triangles and the lengths of two corresponding sides. How do the areas of the two triangles compare? ... more>>

Making a Map - Annie Fetter
Geometry, difficulty level 3. Given a list of relationships between five different objects, explain which ones you could uniquely place on a map. ... more>>

Making a Six Pointed Star - Annie Fetter
Geometry, difficulty level 3. Take a circle of paper. Explain how to fold it so that you can make just one cut and get a "perfect" six-pointed star. ... more>>

More Tangents! - Annie Fetter
Geometry, difficulty level 3. Given a square ABCD inscribed in a unit circle. AB is extended to P, where PC is tangent to the circle. Find the length of PD. ... more>>

My Mom Rocks! - Annie Fetter
Geometry, difficulty level 3. Help my mom decide if she's buried the stones in her new garden deep enough. ... more>>

Nine Congruent Rectangles - Annie Fetter
Geometry, difficulty level 2. Nine congruent rectangles form a larger rectangle whose area is 720 units^2. What's the perimeter of the large rectangle? ... more>>

One Last Bit of Fun - Annie Fetter
Geometry, difficulty level 3. In parallelogram ABCD, the bisector of angle ABC intersects AD at P. If PD=5, BP=6, and CP=6, what is the value of AB? ... more>>

Play Ball! - Annie Fetter
Geometry, difficulty level 2. Figure out how to scale down parts of a basketball court so that it's "woman sized," assuming that the average men's college player is 6'3" and the average woman is 5'6". ... more>>

Pondering Pantographs - Annie Fetter and Lillian Ray
Geometry, difficulty level 3. Why does a pantograph work? ... more>>

Scintillating Similarity - Annie Fetter
Geometry, difficulty level 2. Given a triangle with sides of 3, 4, and 6. What is the perimeter of the smallest triangle that is similar to the first one and has one side with length 12? ... more>>

Seven Congruent Rectangles - Annie Fetter
Geometry, difficulty level 2. Seven congruent rectangles are arranged to form a larger rectangle. If the area of the large rectangle is 336 units^2, what's the perimeter of the large rectangle? ... more>>

Ships Ahoy! - Annie Fetter
Geometry, difficulty level 2. Examine some methods Thales might have used to measure the distance to ships at sea. ... more>>

Similar Perimeter - Annie Fetter
Geometry, difficulty level 2. Find the coordinates of the vertices of a triangle similar to the given triangle with the designated perimeter. ... more>>

So Many Similar Triangles! - Annie Fetter
Geometry, difficulty level 3. Explain how to form pairs of similar triangles given four rods of specific lengths and any other two rods from the extra supply. ... more>>

Square in a Right Triangle - Annie Fetter
Geometry, difficulty level 3. In the given picture, square DEFG is inscribed in a right triangle ABC (with right angle at B) with DG on AC, E on AB, and F on BC. If AD is 14 and GC is 29, what's the area of the square? ... more>>

Squares on Squares - Annie Fetter
Geometry, difficulty level 3. Let P be the center of a 16" x 16" square as well as a vertex of another 16" x 16" square that pivots freely around P. What is the largest possible value for the area where these two squares overlap? ... more>>

A Tangent Square and Circle - Reza Kassai
Geometry, difficulty level 3. The square and circle below are tangent at one point and meet at four other points, as shown in the picture. If the side of the square is 8 units long, what's the radius of the circle? ... more>>

Tangrams - Abram Falk
Geometry, difficulty level 3. Explain which triangles in a tangram are congruent to each other and what the area is of each piece. ... more>>

Two Rectangles - Annie Fetter
Geometry, difficulty level 3. Given the area of one rectangle, find the area of the other rectangle in the picture. ... more>>

Why Does This Construction Work? - Annie Fetter
Geometry, difficulty level 3. Explain why Desmond's construction works. ... more>>

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