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Transformations and Symmetry
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Basic properties of transformations and symmetry are explored in
this section.
Geometry can be thought of as the study of shapes and their transformations.
The problems in this section explore the concepts of reflection, translation,
rotation, and/or dilation, from visual and algebraic perspectives. Since the
high school curriculum does not always address these topics, some of the
problems will provide definitions or explanatory references.
- Related Resources
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Interactive resources from our Math Tools project:
Geometry: Transformations and Symmetry
The closest match in our Ask Dr. Math archives:
High School: Symmetry/Transformations
NCTM Standards:
Geometry Standard for Grades 9-12
Access to these problems requires a Membership.
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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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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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Exploding Shapes
- Annie Fetter
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Geometry, difficulty level 2. How will Clarissa finish her dilation?
... more>>
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A Line of Reflection
- Annie Fetter
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Geometry, difficulty level 2. Find the equation of the line of reflection for triangle ABC and its
image, A'B'C'.
... 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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Mirror, Mirror
- Annie Fetter
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Geometry, difficulty level 2. In the given image, the green polygon is a reflection of the purple
polygon. Find the equation of the line of reflection.
... more>>
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Point P Perambulates
- Annie Fetter
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Geometry, difficulty level 3. Find the length of the path of a vertex of an equilateral triangle as that triangle rotates around the inside of a square.
... more>>
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The Rolling Triangle
- Annie Fetter
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Geometry, difficulty level 3. Equilateral triangle ABC sits on top of unit square DEFG so that AC is
coincident with DE. The triangle is rotated around the outside of the
square until it returns to the top edge. How far did point A travel?
... more>>
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Rotating a Triangle
- Annie Fetter
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Geometry, difficulty level 2. Given a triangle on a coordinate grid, find the new coordinates of the vertices if you rotate the triangle 90 degrees about the origin.
... more>>
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Transforming a Triangle
- Annie Fetter
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Geometry, difficulty level 1. Given two triangles on a coordinate plane, find a pair of transformations that will map the first triangle onto the second triangle.
... more>>
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Triangle in Motion?
- Annie Fetter
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Geometry, difficulty level 2. Given triangle ABC, the triangle is translated and then reflected, and
the locations of A' and A'' are given. Find the locations of B'' and C''.
... more>>
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Triangle Transformation
- Annie Fetter
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Geometry, difficulty level 3. Given triangles ABC and DEF on the coordinate plane and a list of transformations, explain how to map ABC onto DEF in only two steps.
... more>>
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Two Houses and a Carrier Pigeon
- Annie Fetter
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Geometry, difficulty level 3. One house is 300 yards from a highway. 1200 yards down the highway is the driveway to another house, 600 yards from the highway, on the same side of the road. What is the shortest distance between the two houses if you go to the highway in between?
... more>>
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