Lesson 7

Reasoning about Similarity with Transformations

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Problem 1

Sketch a figure that is similar to this figure. Label side and angle measures.

Rhombus. Side lengths 12. Moving clockwise from top left corner, angle measures as follows: 30 degrees, 150 degrees, 30 degrees, 150 degrees.

Solution

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Problem 2

Write 2 different sequences of transformations that would show that triangles \(ABC\) and \(AED\) are similar. The length of \(AC\) is 6 units. 

\(AC=6\)

Triangle A B C and A D E. Point D is located on side A C and point E is to the right of side A C. Side A B is 8, B C is 4, A C is 6, A D is 3, A E is 4, and D E is 2.

Solution

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Problem 3

What is the definition of similarity?

Solution

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(From Unit 3, Lesson 6.)

Problem 4

Select all figures which are similar to Parallelogram \(P\).

Parallelogram \(P\)

Quadrilateral. All sides are 5. Upper left and lower right angles are 50 degrees and upper right and lower left angles are 130 degrees.

Figure \(A\)

Quadrilateral. All sides are 5. Upper left and lower right angles are 50 degrees and upper right and lower left angles are 130 degrees.

Figure \(B\)

Quadrilateral. All sides are 3, left and right side angles are both 50 degrees and upper and lower angles are both 130 degrees.

Figure \(C\)

Quadrilateral. All sides are 5. Upper left and lower right angles are 45 degrees. Upper right and lower left angles are 135 degrees.

Figure \(D\)

Quadrilateral. Left and right side sides are 5, top and bottom sides are 3. Upper left and lower right angles are 50 degrees and upper right and lower left angles are 130 degrees.

Figure \(E\)

Quadrilateral. Bottom side is 10, left side is 5. Base angles are both 50 degrees, upper angles are both 130 degrees.
A:

Figure \(A\)

B:

Figure \(B\)

C:

Figure \(C\)

D:

Figure \(D\)

E:

Figure \(E\)

Solution

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Problem 5

Find a sequence of rigid transformations 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.
A:

Translate by the directed line segment \(AE\), which will take \(B\) to a point \(B’\). Then rotate with center \(E\) by angle \(B’EF\). Finally, dilate with center \(E\) by scale factor \(\frac{5}{2}\).

B:

Translate by the directed line segment \(AE\), which will take \(B\) to a point \(B’\). Then rotate with center \(E\) by angle \(B’EF\). Finally, dilate with center \(E\) by scale factor \(\frac{2}{5}\).

C:

Dilate using center \(E\) by scale factor \(\frac25\).

D:

Dilate using center \(E\) by scale factor \(\frac52\).

Solution

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(From Unit 3, Lesson 6.)

Problem 6

Triangle \(DEF\) is formed by connecting the midpoints of the sides of triangle \(ABC\). What is the perimeter of triangle \(ABC\)? 

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.  

Solution

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(From Unit 3, Lesson 5.)

Problem 7

Select the quadrilateral for which the diagonal is a line of symmetry. 

A:

parallelogram

B:

square

C:

trapezoid

D:

isosceles trapezoid

Solution

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(From Unit 2, Lesson 14.)

Problem 8

Triangles \(FAD\) and \(DCE\) are each translations of triangle \( ABC\)

Large triangle BFE has small triangle ADC at its midpoints.

Explain why angle \(CAD\) has the same measure as angle \(ACB\).

Solution

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(From Unit 1, Lesson 21.)