Notes
Ratios and proportions
- Solve proportions with cross products. Extended ratios $a:b:c$ become $ax$, $bx$, $cx$.
- Geometric mean of $a$ and $b$: $\displaystyle \sqrt{ab}$.
Similar figures
- Corresponding angles congruent, corresponding sides proportional.
- With scale factor $k$: perimeters scale by $k$, areas by $k^2$, volumes by $k^3$.
- A similarity transformation is a dilation, with or without rigid motions.
Proving triangles similar
| Method | What you need |
|---|---|
| AA | 2 pairs of congruent angles |
| SSS | All 3 side ratios equal |
| SAS | 2 side ratios equal and included angles congruent |
Proportional segments
- Triangle Proportionality: a line parallel to a side splits the other two sides proportionally: $\displaystyle \frac{AD}{DB} = \frac{AE}{EC}$.
- Angle Bisector Theorem: $\displaystyle \frac{BD}{DC} = \frac{AB}{AC}$.
Topics in this unit
Each topic has its own short notes and practice.
Practice
Scale factor from $ABC$ to $XYZ$ is $\displaystyle \frac{6}{9} = \frac{2}{3}$, so $\displaystyle YZ = \frac{2}{3} \cdot 12 = 8$.
The area ratio $\displaystyle \frac{32}{72} = \frac{4}{9}$ is $k^2$, so $\displaystyle k = \frac{2}{3}$.
The lamppost's shadow is $12 + 6 = 18$ feet. $\displaystyle \frac{5}{6} = \frac{h}{18}$, so $h = 15$.
$\displaystyle \frac{x}{4} = \frac{6}{8}$, so $x = 3$.
AA. The third choice gives an angle that is not between those sides, so SAS does not apply.
$3.2 \cdot 25 = 80$.
Free response
Write your answer first, then open the scoring guide.
Free-response question
A surveyor uses two similar right triangles to find the width of a river. The small triangle $ABC$ has its right angle at $B$, with leg $AB = 40$ m along the shore and leg $BC = 15$ m. The large triangle $AED$ has its right angle at $E$, with leg $AE = 120$ m along the same line as $\overline{AB}$, and leg $\overline{ED}$ stretching straight across the river, so $ED$ is the river's width. $\triangle ABC \sim \triangle AED$.
(a) Find the scale factor from $\triangle ABC$ to $\triangle AED$.
Show the answer and scoring guide
$\displaystyle \frac{AE}{AB} = \frac{120}{40} = 3$.
Check off the points you earned:
(b) Find the width of the river, $DE$.
Show the answer and scoring guide
$DE = 3 \cdot BC = 3 \cdot 15 = 45$ m.
Check off the points you earned:
(c) The area of $\triangle ABC$ is 300 square meters. Find the area of $\triangle AED$ without using its side lengths, and explain.
Show the answer and scoring guide
Areas scale by $k^2 = 9$, so the area is $300 \cdot 9 = 2700$ square meters.
Check off the points you earned: