Integrated Math 2 / Unit 6: Similarity
6.2 Similar Figures and Proportions
Notes
Two figures are similar ($\sim$) if one is a dilation (possibly with a rigid motion) of the other.
- Corresponding angles are congruent. Corresponding sides are proportional with the same ratio.
- Example: rectangles $4 \times 6$ and $10 \times x$ are similar when $\displaystyle \frac{4}{6} = \frac{10}{x}$, so $4x = 60$ and $x = 15$.
- Match corresponding sides carefully: shortest with shortest, longest with longest.
- Perimeters of similar figures have the same ratio as the sides.
Practice
Triangles with sides $3, 4, 5$ and $9, 12, 15$ are similar. What is the scale factor from the first to the second?
$\displaystyle \frac{9}{3} = \frac{12}{4} = \frac{15}{5} = 3$.
A $4 \times 6$ rectangle is similar to a $10 \times x$ rectangle. Find $x$.
$\displaystyle \frac{4}{6} = \frac{10}{x}$ gives $4x = 60$, so $x = 15$.
Two similar triangles have sides in the ratio $2 : 3$. The smaller triangle has perimeter 20. What is the perimeter of the larger?
Perimeters have the same ratio: $\displaystyle \frac{20}{P} = \frac{2}{3}$, so $P = 30$.
Ready for more? Take the Unit 6 test.