The ratio of the areas of similar triangles

A reminder from eighth grade: two triangles are similar when their corresponding angles are equal and their corresponding sides are in the same ratio.

• In $\displaystyle \triangle ABC\sim\triangle DEF$, the order of the letters says which vertex corresponds to which: $\displaystyle A$ to $\displaystyle D$, $\displaystyle B$ to $\displaystyle E$, and $\displaystyle C$ to $\displaystyle F$.
• The ratio of similarity $\displaystyle k$ is the ratio of corresponding sides: $\displaystyle {\frac{AB}{DE}}={\frac{BC}{EF}}={\frac{CA}{FD}}={k}$.
• The similarity theorems: AA, SAS, SSS.

Theorem: the ratio of the areas of two similar triangles is the square of the ratio of similarity.

$\displaystyle S_1$ and $\displaystyle S_2$ are the areas, and $\displaystyle k$ is the ratio of similarity.

Given: $\displaystyle \triangle ABC\sim\triangle DEF$, with the ratio of similarity $\displaystyle k$.
Prove: $\displaystyle {\frac{S_{ABC}}{S_{DEF}}}={k^2}$.

A triangle’s area is half a side times the height to it. The sides $\displaystyle BC$ and $\displaystyle EF$ are in the ratio $\displaystyle k$. The proof below draws the heights $\displaystyle AH$ and $\displaystyle DK$ and shows that they are in the ratio $\displaystyle k$ too:

From this, the ratio of the areas:

• $\displaystyle {\frac{BC}{EF}}={k}$: the ratio of similarity;
• $\displaystyle {\frac{AH}{DK}}={k}$: the heights are in the same ratio, by steps 5 and 6.

$\displaystyle \triangle ABC\sim\triangle DEF$.
$\displaystyle {AB}={12}$ and $\displaystyle {DE}={4}$.
The area of triangle $\displaystyle DEF$ is $\displaystyle 10$.
What is the area of triangle $\displaystyle ABC$?

• The ratio of similarity: $\displaystyle {k}={\frac{AB}{DE}}={\frac{12}{4}}={3}$;
• the ratio of the areas: $\displaystyle {k^2}={9}$;
• $\displaystyle {S_{ABC}}={9\cdot10}={90}$.

The area of triangle $\displaystyle ABC$ is $\displaystyle 50$, and the area of the similar triangle $\displaystyle DEF$ is $\displaystyle 18$.
The side $\displaystyle AB$ is $\displaystyle 10$.
How long is $\displaystyle DE$?

• The ratio of the areas: $\displaystyle {\frac{50}{18}}={\frac{25}{9}}$;
• the ratio of similarity: $\displaystyle {k}={\sqrt{\frac{25}{9}}}={\frac{5}{3}}$;
• $\displaystyle {\frac{AB}{DE}}={\frac{5}{3}}$, so $\displaystyle {DE}={10\cdot\frac{3}{5}}={6}$.

Calculate. In every part, write the ratio of similarity and the ratio of the areas.

The ratio of similarity of two similar triangles is $\displaystyle 2$, and the area of the smaller triangle is $\displaystyle 7$.
What is the area of the larger one?

The areas of two similar triangles are $\displaystyle 12$ and $\displaystyle 75$.
A side of the smaller triangle is $\displaystyle 4$.
How long is the corresponding side of the larger one?

Every side of one triangle is $\displaystyle 3$ times as long as the corresponding side of another.
How many times as large is the first triangle’s area?

In triangle $\displaystyle ABC$, the point $\displaystyle D$ is on $\displaystyle AB$, the point $\displaystyle E$ is on $\displaystyle AC$, and $\displaystyle DE\parallel BC$.
$\displaystyle {AD}={2}$, $\displaystyle {DB}={4}$, and the area of triangle $\displaystyle ADE$ is $\displaystyle 5$.
Find the area of triangle $\displaystyle ABC$ and the area of the trapezoid $\displaystyle DBCE$.

In triangle $\displaystyle ABC$, the point $\displaystyle D$ is on the side $\displaystyle AB$, the point $\displaystyle E$ is on the side $\displaystyle AC$, and $\displaystyle DE\parallel BC$.

a. Prove: $\displaystyle \triangle ADE\sim\triangle ABC$.

$\displaystyle {AD:AB}={2:5}$, and the area of triangle $\displaystyle ABC$ is $\displaystyle 50$.
Find the area of triangle $\displaystyle ADE$.

In triangle $\displaystyle ABC$, angle $\displaystyle C$ is right, and $\displaystyle CD$ is the height to the hypotenuse.

a. Prove: $\displaystyle \triangle ACD\sim\triangle ABC$.

$\displaystyle {AC}={6}$ and $\displaystyle {AB}={10}$.
Find the area of triangle $\displaystyle ABC$ and the area of triangle $\displaystyle ACD$.