Right triangles and similar triangles

Example:

What is the hypotenuse of a right triangle with leg lengths of $5$ and $12$?

First, remember what each part means:

• The legs are the two shorter sides → $5$ and $12$
• The hypotenuse is the longest side (the side across from the right angle) → this is what we are solving for

We use the Pythagorean theorem:

$$a^2+b^2=c^2$$

Here:

• $a$ and $b$ are the legs
• $c$ is the hypotenuse

Now plug in the values:

$$5^2+12^2=c^2$$

Square each number:

$$25+144=c^2$$

Add them:

$$169=c^2$$

Now we need to undo the square. To do that, take the square root of both sides:

$$\sqrt{169}=\sqrt{c^2}$$

$$13=c$$

The hypotenuse is $13$.

Watch for common whole-number sets like $3$-$4$-$5$, $5$-$12$-$13$, and $8$-$15$-$17$, and their multiples ($6$-$8$-$10$, $9$-$12$-$15$). If two sides fit a pattern, you can read off the third without squaring.

Try one on your own!

Similar triangles

When triangles have three congruent corresponding angles, they are called similar triangles. Their corresponding side lengths are proportional.

Every triangle’s angles add up to $180^{\circ }$, so if two pairs of corresponding angles match, the third pair must match too. Triangles with the same three angles have the same shape: one is just a scaled-up or scaled-down version of the other. Let’s look at how to solve a problem using similar triangles.

Example:

These two triangles are similar. What is the length of $x$?

Images are not to scale

Because the triangles are similar, you can set up a proportion using corresponding sides. The sides labeled $8$ and $x$ correspond, and the sides labeled $15$ and $45$ correspond.

Set up the proportion:

$$\frac{x}{8}=\frac{45}{15}$$

$$\frac{x}{8}=3$$

$$x=24$$

The length of $x$ is $24$.

Special right triangles

These triangles have specific proportional relationships between their angles and side lengths that are useful to memorize.