Lesson 6

Connecting Similarity and Transformations

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Problem 1

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

Squares A B C D and E F G H. A B C D has A upper left, base B C and left side A B is 5. E F G H has G above E, rests on E and right lower side E F is 2.

Solution

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Problem 2

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

  1. What is the scale factor of the dilation that takes \(P\) to \(Q\)?
  2. What is the scale factor of the dilation that takes \(Q\) to \(P\)?
Quadrilaterals P and Q are similar. P has sides from the top measuring 4, 3 and 2, Q has sides from the top 5, none and 2 point 5. 

Solution

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Problem 3

What is our definition of similarity?

A:

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

B:

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

C:

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

D:

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 \(BC\)?

Triangles A B C and D E F. D is the midpoint of segment A B. E is the midpoint of segment B C. F is the midpoint of segment A C. Line D E has length 2, Line E F has length 3, Line D F has line 4.  
A:

3 units

B:

4 units

C:

6 units

D:

8 units

Solution

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(From Unit 3, Lesson 5.)

Problem 5

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

Triangle A B C with segment A’ B’ drawn. A A’ is 2 and A’ C is 4. B B’ is 3 and B’ C is 6. Angle B is 40 degrees.

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

  1. Dilate point \(C\) using center \(D\) and scale factor \(\frac{3}{4}\).
  2. Dilate segment \(AB\) using center \(D\) and scale factor \(\frac12\).
Segments C D and B D intersect at point D. Point A lies on segment B D between B and D.

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? 

A:

\(\frac12\)

B:

\(\frac23\)

C:

\(\frac34\)

D:

\(\frac43\)

Solution

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(From Unit 3, Lesson 3.)

Problem 9

Select all the statements that must be true. 

A:

Parallelograms have four congruent sides.

B:

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

C:

A trapezoid is a parallelogram.

D:

Diagonals of a parallelogram bisect each other. 

E:

Diagonals of a parallelogram are congruent. 

Solution

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(From Unit 2, Lesson 13.)