Right triangles and similar triangles

Triangles — Advanced

In this section, we build on basic triangle concepts to solve more complex problems involving side lengths and proportions.

You will use tools like the Pythagorean theorem, similar triangles, and special right triangle relationships to find missing values and analyze triangle properties.

Pythagorean theorem

Definitions

Hypotenuse: The longest side of the right triangle

The Pythagorean theorem is a formula that lets you find a missing side length in a right triangle (and only a right triangle).

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

In a right triangle:

The shorter sides are called the legs and are labeled $a$ and $b$ . It doesn’t matter which leg is $a$ or which is $b$ — they’re interchangeable.
The hypotenuse (the side opposite the $9 0^{\circ}$ angle) is the longest side and is always labeled $c$ .

A right triangle with legs a and b and hypotenuse c, the side across from the right angle. — Pythagorean theorem

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:

Here:

$a$ and $b$ are the legs

$c$ is the hypotenuse

Now plug in the values:

$5^{2} + 1 2^{2} = c^{2}$

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:

$169 = c^{2}$

$13 = c$

The hypotenuse is $13$ .

Try one on your own!

Knowledge Check

Given a right triangle with a leg length of $3$ inches and a hypotenuse of $5$ inches, what is the length of the second leg?

A. $2$ inches
B. $3$ inches
C. $4$ inches
D. $5$ inches

(spoiler)

C. $4$ inches

Similar triangles

Definitions

Similar triangles: Triangles that are proportional to one another

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

One way to think about it: every triangle’s angles add up to $180$ ° If two triangles have the same three angle measures, then they 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 $x$ ?

$Two similar right triangles. The smaller has legs 15 and 8 and hypotenuse 17. The larger has the matching legs labeled 45 and x, and hypotenuse y.$
Similar triangle problem
Achievable
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 $24$ .

Steps taken:

Identify corresponding sides
Set up a proportion
Simplify the ratio
Solve for the unknown variable
Write the final answer

Special right triangles

There are two types of right triangles that are considered “special” because their angle measures create consistent side-length ratios:

$30$ °- $60$ °- $90$ ° triangles
$45$ °- $45$ °- $90$ ° triangles (also called an isosceles right triangle)

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

A 30-60-90 triangle with sides x opposite the 30° angle, x√3 opposite the 60° angle and hypotenuse 2x, and a 45-45-90 triangle with legs x and x and hypotenuse x√2. — Special right triangles

Example:

What is the hypotenuse of an isosceles right triangle with a leg length of $8$ ?

A. $8$
B. $82$
C. $83$
D. $16$

First, identify the type of triangle.

An isosceles right triangle has:

Two equal sides (the legs)

Two $4 5^{\circ}$ angles

This means it is a $4 5^{\circ}$ - $4 5^{\circ}$ - $9 0^{\circ}$ triangle.

These triangles always follow the same pattern:

$x : x : x 2$

This means:

Each leg = $x$

Hypotenuse = $x 2$

Now plug in what you know.

The leg is $8$ , so:

$x = 8$

Now find the hypotenuse:

$x 2 = 82$

So the hypotenuse is $82$ .

B is the correct answer.

Try one on your own!

Knowledge Check

What is the value of $z$ ?

A right triangle with a 60° angle at the top, a 30° angle at the bottom right and a hypotenuse of 18. The vertical leg is labeled z. — Special triangle example

A. $9$
B. $92$
C. $93$
D. $12$

(spoiler)

A. $9$

Pythagorean theorem

Applies only to right triangles: $a^{2} + b^{2} = c^{2}$
Legs ( $a$ , $b$ ) are shorter sides, interchangeable; hypotenuse ( $c$ ) is longest side, opposite the $90°$ angle
To solve: square legs, add, then take square root to find hypotenuse (or isolate missing leg)

Similar triangles

Definition: triangles with three congruent corresponding angles → proportional corresponding sides
Same angle measures = same shape, different scale
Solving method: identify corresponding sides, set up proportion, cross-multiply/simplify, solve for unknown

Special right triangles

Two key types with fixed side ratios:
- $30°$ - $60°$ - $90°$ : ratio $x : x 3 : 2 x$
- $45°$ - $45°$ - $90°$ (isosceles right triangle): ratio $x : x : x 2$
Memorize ratios to quickly find missing sides once triangle type/angle is identified
For isosceles right triangle: leg = $x$ , hypotenuse = $x 2$

Right triangles and similar triangles

Triangles — Advanced

In this section, we build on basic triangle concepts to solve more complex problems involving side lengths and proportions.

You will use tools like the Pythagorean theorem, similar triangles, and special right triangle relationships to find missing values and analyze triangle properties.

Pythagorean theorem

Definitions

Hypotenuse: The longest side of the right triangle

The Pythagorean theorem is a formula that lets you find a missing side length in a right triangle (and only a right triangle).

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

In a right triangle:

The shorter sides are called the legs and are labeled $a$ and $b$ . It doesn’t matter which leg is $a$ or which is $b$ — they’re interchangeable.
The hypotenuse (the side opposite the $9 0^{\circ}$ angle) is the longest side and is always labeled $c$ .

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:

Here:

$a$ and $b$ are the legs

$c$ is the hypotenuse

Now plug in the values:

$5^{2} + 1 2^{2} = c^{2}$

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:

$169 = c^{2}$

$13 = c$

The hypotenuse is $13$ .

Try one on your own!

Knowledge Check

Given a right triangle with a leg length of $3$ inches and a hypotenuse of $5$ inches, what is the length of the second leg?

A. $2$ inches
B. $3$ inches
C. $4$ inches
D. $5$ inches

(spoiler)

C. $4$ inches

Similar triangles

Definitions

Similar triangles: Triangles that are proportional to one another

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

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 $24$ .

Steps taken:

Identify corresponding sides
Set up a proportion
Simplify the ratio
Solve for the unknown variable
Write the final answer

Special right triangles

There are two types of right triangles that are considered “special” because their angle measures create consistent side-length ratios:

$30$ °- $60$ °- $90$ ° triangles
$45$ °- $45$ °- $90$ ° triangles (also called an isosceles right triangle)

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

Example:

What is the hypotenuse of an isosceles right triangle with a leg length of $8$ ?

A. $8$
B. $82$
C. $83$
D. $16$

First, identify the type of triangle.

An isosceles right triangle has:

Two equal sides (the legs)

Two $4 5^{\circ}$ angles

This means it is a $4 5^{\circ}$ - $4 5^{\circ}$ - $9 0^{\circ}$ triangle.

These triangles always follow the same pattern:

$x : x : x 2$

This means:

Each leg = $x$

Hypotenuse = $x 2$

Now plug in what you know.

The leg is $8$ , so:

$x = 8$

Now find the hypotenuse:

$x 2 = 82$

So the hypotenuse is $82$ .

B is the correct answer.

Try one on your own!

Knowledge Check

What is the value of $z$ ?

A. $9$
B. $92$
C. $93$
D. $12$

(spoiler)

A. $9$