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Exercises: Define Similarity Using Transformations

Work through each section in order. Show your reasoning where indicated. For problems involving figures, identify corresponding parts before checking ratios or angles.

Grade 9·21 problems·~28 min·Common Core Math - HS Geometry·standard·hsg-srt-a-2
Work through problems with immediate feedback
A

Warm-Up: Review What You Know

These problems review skills from earlier lessons.

1.

A dilation centered at the origin maps triangle ABCABC onto triangle A′B′C′A'B'C' with scale factor k=3k = 3. If AB=5AB = 5, what is A′B′A'B'?

2.

Two figures are congruent. Which type of transformation sequence maps one onto the other?

3.

A dilation with scale factor kk is applied to a triangle. Which property is preserved (unchanged) by the dilation?

B

Fluency Practice

1.

A similarity transformation is defined as which sequence of transformations?

2.

Triangle PQRPQR is similar to triangle STUSTU with scale factor k=2.5k = 2.5. If PQ=8PQ = 8, what is the length of STST?

3.

Triangle ABCABC has sides AB=6AB = 6, BC=8BC = 8, CA=10CA = 10. Triangle DEFDEF has sides DE=9DE = 9, EF=12EF = 12, FD=15FD = 15. Are the triangles similar?

4.

Rectangle ABCDABCD has dimensions 4×104 \times 10. Rectangle EFGHEFGH has dimensions 6×156 \times 15. What is the scale factor kk from ABCDABCD to EFGHEFGH?

5.

Triangle JKL∼JKL \sim triangle MNPMNP with scale factor k=3k = 3. If ∠J=47∘\angle J = 47^\circ and ∠K=68∘\angle K = 68^\circ, what is ∠N\angle N?

C

Varied Practice

1.

If a scale factor of k=1k = 1 is used in a similarity transformation, what is the result?

2.

Triangle ABCABC has angles 60∘60^\circ, 80∘80^\circ, and 40∘40^\circ and is oriented upright. Triangle DEFDEF has the same three angle measures but is flipped (reflected). Can a similarity transformation map △ABC\triangle ABC onto △DEF\triangle DEF?

3.

Pentagon ABCDEABCDE has sides 3,4,5,6,73, 4, 5, 6, 7 (in order). Pentagon FGHIJFGHIJ has sides 7.5,10,12.5,15,17.57.5, 10, 12.5, 15, 17.5 (in order). The pentagons are similar with corresponding sides in the same order. What is the scale factor kk from ABCDEABCDE to FGHIJFGHIJ?

4.

Triangle RSTRST has sides 5,12,135, 12, 13 and angles 23∘,67∘,90∘23^\circ, 67^\circ, 90^\circ.
Triangle UVWUVW has sides 5,12,135, 12, 13 and angles 23∘,67∘,90∘23^\circ, 67^\circ, 90^\circ.
Which statement best describes these two triangles?

5.

In everyday language, people say two things are "similar" when they look approximately alike. Explain how the mathematical definition of similarity is more precise, and give one example that shows the difference.

D

Word Problems

1.

At noon on a sunny day, a flagpole casts a shadow of 15 m. At the same moment, a nearby student who is 1.6 m tall casts a shadow of 2.4 m. The two right triangles formed by the heights and shadow lengths are similar (same sun angle).

1.

Use the similarity of the triangles to find the height hh of the flagpole in meters.

2.

What is the scale factor kk from the student triangle to the flagpole triangle?

2.

An architect creates a scale model of a building. The model is 0.8 m tall and 1.2 m wide. The actual building is 32 m tall. The model and the building are similar figures.

How wide is the actual building, in meters?

3.

Mia claims that all squares are similar to each other. Her reasoning: "Any square can be mapped onto any other square using a translation and a dilation — no rotation or reflection is ever needed."

Is Mia's claim correct? Choose the best response.

E

Error Analysis

1.

Jordan is asked whether the two triangles below are similar.

Triangle PQRPQR: sides 6,8,106, 8, 10.
Triangle STUSTU: sides 6,8,106, 8, 10.

Jordan writes: "The triangles are similar because their sides are proportional with k=1k = 1. But they are NOT congruent because similar figures always have different sizes."

What error did Jordan make?

2.

Priya is told that two quadrilaterals "look similar" in her textbook's glossary, which says figures are similar if they "resemble each other in shape." She concludes that a 3×43 \times 4 rectangle and a 3×53 \times 5 rectangle are similar because both are rectangles that "look alike."

Explain the flaw in Priya's reasoning and state the correct mathematical test.

F

Challenge

1.

Triangle ABCABC is similar to triangle DEFDEF with scale factor kk. The perimeter of △ABC\triangle ABC is 30 and the perimeter of △DEF\triangle DEF is 45. If side AB=8AB = 8, what is the length of side DEDE?

2.

A student claims: "Two triangles must be similar if they have two pairs of equal corresponding angles." Using the definition of similarity in terms of transformations, explain why this claim is correct. Your response should reference what happens to all three angles and all three sides when a similarity transformation is applied.

0 of 21 answered