Are They All Similar?

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Created by Illustrative MathematicsIllustrative Mathematics, CC BY 4.0
Subject: Geometry
Grade: 9-12
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9 questions

Are They All Similar?

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Problem 1

1) This is an invalid proof that all isosceles triangles are similar. Explain which step is invalid and why.

1. Draw 2 isosceles triangles ABCABC and DEFDEF where AC=BCAC = BC and DF=EFDF = EF.

2. Dilate triangle ABCABC to a new triangle ABCA'B'C using center CC and scale factor DFAC\frac{DF}{AC} so that AC=BC=DF=EFA'C = B'C = DF = EF.

3. Translate by directed line segment CFCF to take ABCA'B'C to a new triangle ABFA''B''F. Since translation preserves distance, AF=AC=DFA''F = A'C = DF and BF=BC=EFB''F = B'C = EF.

4. Since AF=DFA''F = DF, we can rotate using center FF to take AA'' to DD.

5. Since BF=EFB''F = EF, we can rotate using center FF to take BB'' to EE.

6. We have now established a sequence of dilations, translations, and rotations that takes AA to DD, BB to EE, and CC to FF, so the triangles are similar.

Problem 2

2) Which statement provides a valid justification for why all circles are similar?

a)

All circles have the same shape—a circle—so they must be similar.

b)

All circles have no angles and no sides, so they must be similar.

c)

I can translate any circle exactly onto another, so they must be similar.

d)

I can translate the center of any circle to the center of another, and then dilate from that center by an

appropriate scale factor, so they must be similar.

Problem 3

3) Which pair of polygons is similar?

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a)

Figure A

b)

Figure B

c)

Figure C

d)

Figure D

Problem 4

4) Select all\textbf{all} sequences of transformations that would show that triangles ABCABC and AEDAED are similar. The length of ACAC is 6 units. Write each corresponding letter in the answer box and separate letters with commas.

a) Dilate triangle ABCABC using center AA by a scale factor of 12\frac{1}{2}, then reflect over line ACAC.

b) Dilate triangle AEDAED using center AA by a scale factor of 2, then reflect over line ACAC.

c) Reflect triangle ABCABC over line ACAC, then dilate using center AA by a scale factor of 12\frac{1}{2}.

d) Reflect triangle AEDAED over line ACAC, then dilate using center AA by a scale factor of 2.

e) Translate triangle AEDAED by directed line segment DCDC, then dilate using center CC by scale factor 2.

f) Translate either triangle ABCABC or AEDAED by directed line segment DCDC, then reflect over line ACAC.

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Problem 5

Determine if each statement must be true, could possibly be true, or definitely can't be true.

5) Two equilateral triangles are similar.

a)

Must be true

b)

Could be true

c)

Cannot be true

6) An equilateral triangle and a square are similar.

a)

Must be true

b)

Could be true

c)

Cannot be true

7) Explain or show your reasoning for each of your responses.

Problem 6

8) Find a sequence of rigid transformations and dilations that takes square EFGHEFGH to square ABCDABCD.

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Problem 7

9) Select all\textbf{all} true statements. Write each corresponding letter in the answer box and separate letters with commas.

a) Angle ACBACB is 180x°180 - x\degree \quad\quad b) Angle ACBACB is x°x\degree \quad\quad c) Triangle ACBACB is similar to triangle ADEADE

d) AD=13ADAD = \frac{1}{3}AD \quad\quad e) AD=12DCAD = \frac{1}{2}DC

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