TOPICS
This page:
similarity, congruence
About Levels
of Difficulty
Geometry
lines/angles
triangles
quadrilaterals
polygons
circles
perimeter, area,
volume
similar/congruent
right triangles
transform/symmetry
construction/locus
coordinate geometry
modeling/applications
Browse all
Geometry
Problems
Geometry
Home
About the
PoW Library


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 912
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 15inchdeep slice off an 8.5foot 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/(1x) = 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 XGames
 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" sixpointed 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>>
Page: 1
