Similar figures and similar triangles

Enlargement and scale factor, the four tests for similar triangles, matching sides and finding unknown lengths.

Scale factor

$$k = \frac{\text{image length}}{\text{original length}}$$

Similar figures have equal matching angles and matching sides in the same ratio. We write $\triangle ABC \sim \triangle DEF$.

Example 1●●

$\triangle ABC \sim \triangle DEF$ with $AB = 5$, $DE = 15$ and $BC = 7$. Find $EF$.

$k = 15 \div 5 = 3$

$EF = 7 \times 3 = 21$

Example 2●●

Are triangles with sides $4, 6, 8$ and $6, 9, 12$ similar?

$\dfrac{6}{4} = \dfrac{9}{6} = \dfrac{12}{8} = 1.5$

Yes, by SSS (sides in the same ratio), with $k = 1.5$.

Flashcards

$sim$

is similar to

Similarity test using angles

AAA (two angles is enough)

Matching sides

Opposite equal angles

Practice

$\triangle ABC \sim \triangle DEF$, $AB = 5$, $DE = 15$, $BC = 7$. Find $EF$.
Triangles with sides $4, 6, 8$ and $6, 9, 12$. Scale factor?
Two triangles have angles $50^\circ, 60^\circ, 70^\circ$. They are:
Are triangles with sides $3, 4, 5$ and $6, 8, 9$ similar?
$\triangle PQR \sim \triangle XYZ$ with $PQ = 6$, $XY = 3$, $QR = 8$. Find $YZ$.
A triangle with base $4$ and height $3$ is similar to one with base $12$. Height of the larger?
A $9$ cm side matches a $6$ cm side in the image. Scale factor (original to image) as a fraction?
Which pair of rectangles is similar?