# Lesson 6

Connecting Similarity and Transformations

### Problem 1

Find a sequence of rigid motions and dilations that takes square \(ABCD\) to square \(EFGH\).

### Solution

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### Problem 2

Quadrilaterals \(Q\) and \(P\) are similar.

- What is the scale factor of the dilation that takes \(P\) to \(Q\)?
- What is the scale factor of the dilation that takes \(Q\) to \(P\)?

### Solution

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### Problem 3

What is our definition of similarity?

If 2 figures have the same angles, then they are similar.

If 2 figures have proportional side lengths, then they are similar.

If there is a sequence of rigid transformations taking one figure to another, then they are similar.

If there is a sequence of rigid transformations and dilations that take one figure to the other, then they are similar.

### Solution

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### Problem 4

Triangle \(DEF\) is formed by connecting the midpoints of the sides of triangle \(ABC\). The lengths of the sides of \(DEF\) are shown. What is the length of \(BC\)?

3 units

4 units

6 units

8 units

### Solution

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(From Unit 3, Lesson 5.)### Problem 5

If \(AB\) is 12, what is the length of \(A'B'\)?

### Solution

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(From Unit 3, Lesson 5.)### Problem 6

Right angle \(ABC\) is taken by a dilation with center \(P\) and scale factor \(\frac12\) to angle \(A’B’C’\). What is the measure of angle \(A'B'C'\)?

### Solution

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(From Unit 3, Lesson 4.)### Problem 7

- Dilate point \(C\) using center \(D\) and scale factor \(\frac{3}{4}\).
- Dilate segment \(AB\) using center \(D\) and scale factor \(\frac12\).

### Solution

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(From Unit 3, Lesson 4.)### Problem 8

A polygon has perimeter 12. It is dilated with a scale factor of \(k\) and the resulting image has a perimeter of 8. What is the scale factor?

\(\frac12\)

\(\frac23\)

\(\frac34\)

\(\frac43\)

### Solution

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(From Unit 3, Lesson 3.)### Problem 9

Select **all** the statements that *must*** **be true.

Parallelograms have four congruent sides.

Both sets of opposite sides of a parallelogram are parallel and congruent.

A trapezoid is a parallelogram.

Diagonals of a parallelogram bisect each other.

Diagonals of a parallelogram are congruent.

### Solution

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(From Unit 2, Lesson 13.)