Similar triangles

Two shapes are similar when one of them is an enlargement or a reduction of the other: they have the same shape, and their size may differ.

• When you zoom into a photo on a phone with two fingers, every length in it grows by the same number, and the photo is not distorted.
• A map is a reduction: every distance on it is smaller by the same number – the scale.
• Congruent shapes are similar too: one is an “enlargement by $\displaystyle 1$” of the other.

Two triangles are similar when their corresponding angles are equal and their corresponding sides are in the same ratio. We write $\displaystyle \triangle ABC\sim\triangle DEF$ and read: “triangle $\displaystyle ABC$ is similar to triangle $\displaystyle DEF$”.

• The corresponding angles are equal: $\displaystyle {\angle A}={\angle D}$, $\displaystyle {\angle B}={\angle E}$, $\displaystyle {\angle C}={\angle F}$.
• The corresponding sides are in the same ratio: $\displaystyle {\frac{DE}{AB}}={\frac{EF}{BC}}={\frac{FD}{CA}}$. In the drawing: $\displaystyle {\frac{6}{4}}={\frac{9}{6}}={\frac{7.5}{5}}={1.5}$.
• The order of the letters says which corresponds to which, as in congruence: $\displaystyle A$ to $\displaystyle D$, $\displaystyle B$ to $\displaystyle E$, and $\displaystyle C$ to $\displaystyle F$.
• A side corresponds to the side between the corresponding vertices: $\displaystyle AB$ to $\displaystyle DE$, and $\displaystyle BC$ to $\displaystyle EF$.

The ratio of similarity is the number you multiply each side of one triangle by to get the corresponding side of the other. It is written $\displaystyle k$.

• In the drawing $\displaystyle {k}={1.5}$: each side of $\displaystyle DEF$ is $\displaystyle 1.5$ times the corresponding side of $\displaystyle ABC$. Triangle $\displaystyle DEF$ is an enlargement of $\displaystyle ABC$.
• The other way, from $\displaystyle DEF$ to $\displaystyle ABC$, the ratio of similarity is $\displaystyle {\frac{4}{6}}={\frac{2}{3}}$: a reduction.
• When the ratio of similarity is $\displaystyle 1$, the triangles are congruent.

Given: $\displaystyle \triangle ABC\sim\triangle KLM$.

Write the three pairs of corresponding angles and the three pairs of corresponding sides.

Also given: $\displaystyle {AB}={5}$ and $\displaystyle {KL}={15}$.
What is the ratio of similarity from $\displaystyle ABC$ to $\displaystyle KLM$? And from $\displaystyle KLM$ to $\displaystyle ABC$?

Given: $\displaystyle \triangle ABC\sim\triangle DEF$.
The lengths: $\displaystyle {AB}={6}$, $\displaystyle {BC}={8}$, $\displaystyle {DE}={9}$.
How long is $\displaystyle EF$?

The solution below finds the ratio of similarity from a pair of corresponding sides whose lengths are known, and multiplies $\displaystyle BC$ by it.

• $\displaystyle AB$ corresponds to $\displaystyle DE$: both are between the first vertex and the second. The ratio of similarity: $\displaystyle {k}={\frac{DE}{AB}}={\frac{9}{6}}={1.5}$.
• $\displaystyle EF$ corresponds to $\displaystyle BC$, so $\displaystyle {EF}={1.5\cdot8}={12}$.

A proportion works too: $\displaystyle {\frac{EF}{8}}={\frac{9}{6}}$. Cross-multiplying, we get $\displaystyle {6\cdot EF}={72}$, that is $\displaystyle {EF}={12}$.

Given: $\displaystyle \triangle ABC\sim\triangle DEF$.
The lengths: $\displaystyle {AB}={12}$, $\displaystyle {BC}={9}$, $\displaystyle {CA}={15}$, $\displaystyle {DE}={8}$.
In each part, write a proportion and solve it.

Find $\displaystyle EF$.

Find $\displaystyle FD$.

Given: $\displaystyle \triangle PQR\sim\triangle XYZ$.
The lengths: $\displaystyle {PQ}={4}$, $\displaystyle {XY}={10}$, $\displaystyle {YZ}={15}$.
Find $\displaystyle QR$.

In an enlargement by $\displaystyle k$, the perimeter grows by $\displaystyle k$ too: each side grows by $\displaystyle k$, and so does their sum.

• In the triangles of the drawing, the perimeter of $\displaystyle ABC$ is $\displaystyle {4+6+5}={15}$, and the perimeter of $\displaystyle DEF$ is $\displaystyle {6+9+7.5}={22.5}$.
• $\displaystyle {15\cdot1.5}={22.5}$: the perimeter grew $\displaystyle 1.5$ times, like the sides.

The triangle with sides $\displaystyle 3$, $\displaystyle 4$ and $\displaystyle 5$ is enlarged by $\displaystyle 3$.

How long are the sides of the new triangle?

What is the perimeter of each of the two triangles? How many times larger is the new perimeter?

One equilateral triangle has sides of $\displaystyle 4$, another has sides of $\displaystyle 10$.
Are the triangles similar? If so, what is the ratio of similarity?
Are any two equilateral triangles similar? Explain.