# Are They All Similar?

ID: nunaf-gusih
Illustrative Math
Subject: Geometry

# 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 ﻿$ABC$﻿ and ﻿$DEF$﻿ where ﻿$AC = BC$﻿ and ﻿$DF = EF$﻿.

2. Dilate triangle ﻿$ABC$﻿ to a new triangle ﻿$A'B'C$﻿ using center ﻿$C$﻿ and scale factor ﻿$\frac{DF}{AC}$﻿ so that ﻿$A'C = B'C = DF = EF$﻿.

3. Translate by directed line segment ﻿$CF$﻿ to take ﻿$A'B'C$﻿ to a new triangle ﻿$A''B''F$﻿. Since translation preserves distance, ﻿$A''F = A'C = DF$﻿ and ﻿$B''F = B'C = EF$﻿.

4. Since ﻿$A''F = DF$﻿, we can rotate using center ﻿$F$﻿ to take ﻿$A''$﻿ to ﻿$D$﻿.

5. Since ﻿$B''F = EF$﻿, we can rotate using center ﻿$F$﻿ to take ﻿$B''$﻿ to ﻿$E$﻿.

6. We have now established a sequence of dilations, translations, and rotations that takes ﻿$A$﻿ to ﻿$D$﻿, ﻿$B$﻿ to ﻿$E$﻿, and ﻿$C$﻿ to ﻿$F$﻿, so the triangles are similar.

##### Problem 2

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

a) $\text{All circles have the same shape—a circle—so they must be similar.}$b) $\text{All circles have no angles and no sides, so they must be similar.}$c) $\text{I can translate any circle exactly onto another, so they must be similar.}$d) $\text{I can translate the center of any circle to the center of another, and then dilate from that center by an } \newline \text{ appropriate scale factor, so they must be similar.}$
##### Problem 3

3) Which pair of polygons is similar?

a) $\text{Figure A}$b) $\text{Figure B}$c) $\text{Figure C}$d) $\text{Figure D}$
##### Problem 4

4) Select all sequences of transformations that would show that triangles ﻿$ABC$﻿ and ﻿$AED$﻿ are similar. The length of ﻿$AC$﻿ is 6 units. Write each corresponding letter in the answer box and separate letters with commas.

a) Dilate triangle ﻿$ABC$﻿ using center ﻿$A$﻿ by a scale factor of ﻿$\frac{1}{2}$﻿, then reflect over line ﻿$AC$﻿.

b) Dilate triangle ﻿$AED$﻿ using center ﻿$A$﻿ by a scale factor of 2, then reflect over line ﻿$AC$﻿.

c) Reflect triangle ﻿$ABC$﻿ over line ﻿$AC$﻿, then dilate using center ﻿$A$﻿ by a scale factor of ﻿$\frac{1}{2}$﻿.

d) Reflect triangle ﻿$AED$﻿ over line ﻿$AC$﻿, then dilate using center ﻿$A$﻿ by a scale factor of 2.

e) Translate triangle ﻿$AED$﻿ by directed line segment ﻿$DC$﻿, then dilate using center ﻿$C$﻿ by scale factor 2.

f) Translate either triangle ﻿$ABC$﻿ or ﻿$AED$﻿ by directed line segment ﻿$DC$﻿, then reflect over line ﻿$AC$﻿.

##### 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) $\text{Must be true}$b) $\text{Could be true}$c) $\text{Cannot be true}$

6) An equilateral triangle and a square are similar.

a) $\text{Must be true}$b) $\text{Could be true}$c) $\text{Cannot be true}$

8) Find a sequence of rigid transformations and dilations that takes square ﻿$EFGH$﻿ to square ﻿$ABCD$﻿.
a) Angle ﻿$ACB$﻿ is ﻿$180 - x\degree$﻿ ﻿$\quad\quad$﻿ b) Angle ﻿$ACB$﻿ is ﻿$x\degree$﻿ ﻿$\quad\quad$﻿ c) Triangle ﻿$ACB$﻿ is similar to triangle ﻿$ADE$﻿
d) ﻿$AD = \frac{1}{3}AD$﻿ ﻿$\quad\quad$﻿ e) ﻿$AD = \frac{1}{2}DC$﻿