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 similarity, congruence
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Similarity and Congruence
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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
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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.
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Arda and Brooke's Catapult
- Annie Fetter
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Geometry, difficulty level 2. Does Arda's method for measuring the distance across the river really
work?
... more>>
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Area of a Rectangle
- Annie Fetter
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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>>
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Building a Bridge Across a Chasm
- Annie Fetter
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Geometry, difficulty level 2. So they're standing on the edge of this chasm. It isn't really that far across, but they need to find a log or tie together enough rope to reach a tree on the other side. So they mark a spot on the edge of the chasm directly across from the tree. Then they walk 20 paces along the chasm, and she stays there. He walks another 20 paces and marks the spot. Then he turns ninety degrees and walks away from the chasm until she's directly in the line of sight between him and the tree on the other side. Then he claims that the distance he just walked from the edge of the chasm is the same as the distance across the chasm.
... more>>
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Building a Vaulted Ceiling
- Annie Fetter
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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>>
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A Carpenter's Trisection
- Annie Fetter
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Geometry, difficulty level 3. Explain whether or not the given method of trisecting an angle, using only a carpenter's square, really works.
... more>>
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A Circle Inscribed in an Isosceles Triangle
- Ilmar Vitsut
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Geometry, difficulty level 4. Find the radius of a circle inscribed in an isosceles triangle with sides 12, 12, and 8.
... more>>
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Circles and Tangents
- Annie Fetter
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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>>
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Clever Cuts
- Annie Fetter
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Geometry, difficulty level 2. Find three different ways to cut a hexagonal cake into six congruent
pieces.
... more>>
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Congruence and Area
- Annie Fetter
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Geometry, difficulty level 3. "If two triangles have the same area, then they are congruent." Is this a true statement?
... more>>
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Congruent Chords and Circles
- Annie Fetter
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Geometry, difficulty level 4. Prove that two congruent chords in a circle are equal distances from the center of the circle.
... more>>
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Congruent Rectangles
- Annie Fetter
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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>>
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Constructing the Great Pyramid
- Annie Fetter & Suzanne Alejandre
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Geometry, difficulty level 2. Figure out if the instructions provided will result in a scale
model of the Great Pyramid of Giza.
... more>>
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The Diagonal Triangle
- Annie Fetter
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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>>
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Dilating a Quadrilateral
- Annie Fetter
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Geometry, difficulty level 3. Given two similar quadrilaterals on a coordinate grid, determine the ratio and center of dilation.
... more>>
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The Dimensions of a Hunting Platform
- Annie Fetter
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Geometry, difficulty level 2. I have a picture of a hunting platform, and I need help figuring out some of the dimensions.
... more>>
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Drawing a Kite Plan
- Annie Fetter
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Geometry, difficulty level 3. Given a picture of a kite plan, figure out the radius of the circle that creates the curved section.
... more>>
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Equal Area & Congruence
- Annie Fetter
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Geometry, difficulty level 2. If two triangles are congruent, they have the same area. Is the converse true?
... more>>
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Equilateral Triangle and Square Ratio
- Dale Pearson
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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>>
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Explanation Only!
- Annie Fetter
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Geometry, difficulty level 4. Prove that x/(1-x) = 1/x in the given picture.
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Finding CE
- Annie Fetter
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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?
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Finding the Side of a Parallelogram
- Annie Fetter
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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>>
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Folding a Piece of Paper in Thirds
- Annie Fetter
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Geometry, difficulty level 3. Show that the given folds mark a piece of paper exactly in thirds.
... more>>
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Geometry X-Games
- Annie Fetter
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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>>
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An Inscribed Circle
- Annie Fetter
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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>>
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Kyle's Similar Triangles
- Kyle Binder and Annie Fetter
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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>>
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Making a Map
- Annie Fetter
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Geometry, difficulty level 3. Given a list of relationships between five different objects, explain which ones you could uniquely place on a map.
... more>>
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Making a Six Pointed Star
- Annie Fetter
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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>>
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More Tangents!
- Annie Fetter
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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>>
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My Mom Rocks!
- Annie Fetter
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Geometry, difficulty level 3. Help my mom decide if she's buried the stones in her new garden deep enough.
... more>>
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Nine Congruent Rectangles
- Annie Fetter
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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>>
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One Last Bit of Fun
- Annie Fetter
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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>>
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Play Ball!
- Annie Fetter
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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>>
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Scintillating Similarity
- Annie Fetter
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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>>
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Seven Congruent Rectangles
- Annie Fetter
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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>>
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So Many Similar Triangles!
- Annie Fetter
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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.
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Square in a Right Triangle
- Annie Fetter
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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>>
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Squares on Squares
- Annie Fetter
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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?
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A Tangent Square and Circle
- Reza Kassai
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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>>
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Tangrams
- Abram Falk
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Geometry, difficulty level 3. Explain which triangles in a tangram are congruent to each other and what the area is of each piece.
... more>>
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Tangrams, Congruent Triangles, and Area
- Annie Fetter
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Geometry, difficulty level 3. Tell me which pieces of a tangram are congruent, and tell me the area of each of the pieces.
... more>>
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Triangles, Area, and Congruence
- Annie Fetter
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Geometry, difficulty level 2. If two triangles have equal area, does that mean that they are congruent?
... more>>
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