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Integrated Math 2

Unit 6

Similarity

7 topics · 27 practice questions · Free responseNot yet reviewed

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Notes

Dilations and similar figures

  • A dilation with scale factor $k$ about the origin sends $(x, y)$ to $(kx, ky)$. It keeps angles and multiplies lengths by $k$.
  • Similar figures have congruent corresponding angles and proportional corresponding sides. Set up proportions with matching sides.

Proving triangles similar

  • AA, SSS (all sides proportional) and SAS (two sides proportional, included angle congruent). SSA does not work.
  • Useful angle facts: vertical angles, alternate interior and corresponding angles with parallel lines, a shared angle.

Using similarity

  • If $DE \parallel BC$ in triangle $ABC$, then $\displaystyle \frac{AD}{DB} = \frac{AE}{EC}$. A midsegment is parallel to the third side and half as long.
  • Shadow problems: set height over shadow equal for both objects.
  • Scale factor $k$: perimeter ratio $k$, area ratio $k^2$.

Topics in this unit

Each topic has its own short notes and practice.

Practice

A dilation about the origin with scale factor 2 is applied to $(3, -1)$. What is the image?
$\triangle ABC \sim \triangle DEF$ with $AB = 6$, $DE = 9$ and $BC = 10$. Find $EF$.
Which condition is NOT enough to prove two triangles similar?
A 5 ft tall child casts an 8 ft shadow. At the same time a flagpole casts a 40 ft shadow. How tall is the flagpole?
Two similar figures have sides in the ratio $2 : 5$. If the smaller has area 12, what is the area of the larger?
A triangle has sides 10, 14 and 18. A midsegment is parallel to the side of length 14. What is its length?

Free response

Write your answer first, then open the scoring guide.

Free-response question

In triangle $ABC$, point $D$ is on $AB$ and point $E$ is on $AC$ with $DE \parallel BC$. Given $AD = 6$, $DB = 3$ and $DE = 8$.

(a) Prove that $\triangle ADE \sim \triangle ABC$.

Show the answer and scoring guide

$\angle A \cong \angle A$ (shared angle). $\angle ADE \cong \angle ABC$ (corresponding angles, since $DE \parallel BC$). So the triangles are similar by AA.

Check off the points you earned:

(b) Find $BC$.

Show the answer and scoring guide

$AB = 6 + 3 = 9$. The scale factor from $\triangle ADE$ to $\triangle ABC$ is $\displaystyle \frac{9}{6} = 1.5$. $BC = 8 \cdot 1.5 = 12$.

Check off the points you earned:

(c) If the area of $\triangle ADE$ is 24, find the area of $\triangle ABC$.

Show the answer and scoring guide

The ratio of areas is $\displaystyle \left(\frac{6}{9}\right)^2 = \frac{4}{9}$. So $\displaystyle \frac{24}{A} = \frac{4}{9}$ and $A = 54$.

Check off the points you earned:

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