Lesson 8

Are They All Similar?

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Problem 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.

Solution

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Problem 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.

Solution

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Problem 3

Which pair of polygons is similar? 

A:
Right triangle with side lengths 5, 12, 13.
Right triangle with side lengths 3, 4, 5.  
B:
Quadrilateral. 4 right angles. Each side length 4.  
Quadrilateral. Each side length 4.  
C:
Quadrilateral. 4 right angles. Side lengths 1, 5, 1, 5.  
Quadrilateral. 4 right angles. Side lengths 2, 10, 2, 10.  
D:
riangle. Base length, 6. Side lengths, 10.  
Triangle. Base length, 6. Side lengths, 5.  

Solution

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Problem 4

Select all 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.
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 \(AC\).

Solution

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(From Unit 3, Lesson 7.)

Problem 5

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

  1. Two equilateral triangles are similar.
  2. An equilateral triangle and a square are similar.

Solution

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(From Unit 3, Lesson 7.)

Problem 6

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

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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(From Unit 3, Lesson 6.)

Problem 7

Select all true statements given that angle \(AED\) is congruent to angle \(ABC\)

Triangle A B C. Point E on Side A B, Point D on Side C A, line segment E D drawn, parallel to B C, labeled 2. Segment A E labeled 5. Segment E B labeled 10. Side B C labeled 6. Angle E D A labeled x degrees.
A:

Angle \(ACB\) is \(180-x^\circ \)

B:

Angle \(ACB\) is \(x^{\circ}\)

C:

Triangle \(ACB\) is similar to triangle \(ADE\)

D:

\(AD = \frac{1}{3}AC\)

E:

\(AD = \frac{1}{2}DC\)

Solution

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(From Unit 3, Lesson 5.)