Lesson 6

Connecting Similarity and Transformations

  • Let’s identify similar figures.

6.1: Dilation Miscalculation

Two diagonal line segments intersect at point P. Between two line segments, Quadrilaterals G H F E and A D C B, both with 4 right angles. G H F E looks like a rectangle, A D C B looks like a square.

What’s wrong with this dilation? Why is \(GHFE\) not a dilation of \(ADCB\)?

6.2: Card Sort: Not-So-Rigid Transformations

  1. Your teacher will give you a set of cards. Sort the cards into categories of your choosing. Be prepared to explain the meaning of your categories.
  2. Your teacher will assign you one card. Write the sequence of transformations (translation, rotation, reflection, dilation) to take one figure to the other.
  3. For all the cards that could include a dilation, what scale factor is used to go from Figure \(F\) to Figure \(G\)? What scale factor is used to go from Figure \(G\) to Figure \(F\)?


Find a sequence of transformations that takes Figure \(G\) to Figure \(F\). How does this sequence compare to the sequence that took Figure \(F\) to Figure \(G\)?

6.3: Alphabet Soup

Are the triangles similar?

\(\overline{AB}\parallel\overline{QR},\overline{AB}\perp\overline{AE},\overline{QR}\perp\overline{QT}\)

Right Triangles A B E & Q R T. A B, 8. A E, 15. B E, 17. Angle B A E, right. Q R, 3 point 2. Q T, 6. R T, 6 point 8. Angle R Q T, right. Angles A E B & Q T R, congruent. Angles A B E & Q R T, congruent.
  1. Write a sequence of transformations (dilation, translation, rotation, reflection) to take one triangle to the other.
  2. Write a similarity statement about the 2 figures, and explain how you know they are similar.
  3. Compare your statement with your partner’s statement. Is there more than one correct way to write a similarity statement? Is there a wrong way to write a similarity statement?

Summary

One figure is similar to another if there is a sequence of rigid motions and dilations that takes the first figure so that it fits exactly over the second. For example, triangle \(DHF\) is similar to triangle \(EHG\). What is a rotation and a dilation that will take \(DHF\) onto \(EHG\)?

Two parallel horizontal line segments. Top, F D, bottom, E G, point H between two segments. Triangles D H F and E H G drawn. D H labeled 2, H F labeled 2 point 8. H G labeled 5 point 6, H E labeled 4.

The triangles are similar because a \(180^{\circ}\) rotation of \(DHF\) using center \(H\) will take segment \(HF\) onto segment \(HG\), since \(180^{\circ}\) rotations take lines through the center of the rotation to themselves. It will also take \(HD\) onto \(HE\) for the same reason. Then \(G\) will be on a ray from \(H\) through \(F’\), and \(E\) will be on a ray from \(H\) through \(D’\). Since \(\frac{H’F’}{HG} = \frac{H’D’}{HE} = \frac12\), a dilation by a scale factor of 2 will take \(D’H’F’\) onto \(EHG\), which means there is a sequence of rigid motions and dilations that takes \(DHF\) onto \(EHG\). 

Adding on from previous image. Triangles D H F, blue, and E H G drawn. Point D prime on H E. Point F prime on H G. Triangle D prime H prime F prime drawn, green.

Since similar figures are the result of rigid motions and dilations, in similar figures, all pairs of corresponding angles are congruent, and the lengths of all pairs of corresponding sides are in the same proportion. Angle \(D\) is congruent to angle \(E\). Angle \(F\) is congruent to angle \(G\). Angle \(DHF\) is congruent to angle \(EHG\). And \(\frac{HD}{HE}\)=\(\frac{HF}{HG}\)=\(\frac{DF}{EG}\).

We use \(\sim\) as a symbol for is similar to, so we read \(\triangle DHF \sim \triangle EHG\) as “triangle \(DHF\) is similar to triangle \(EHG\).”

Glossary Entries

  • similar

    One figure is similar to another if there is a sequence of rigid motions and dilations that takes the first figure onto the second.

    Triangle \(A'B'C'\) is similar to triangle \(ABC\) because a rotation with center \(B\) followed by a dilation with center \(P\) takes \(ABC\) to \(A'B'C'\).

    Triangle ABC, rotated and then dilated.