While the Pythagorean theorem can apply to any kind of right triangle, there are particular types of right triangles whose side lengths and angles have helpful properties.

45\degree-45\degree-90\degree triangle theorem

In a 45\degree-45\degree-90\degree triangle, the legs are congruent and the length of the hypotenuse is \sqrt{2} times the length of the legs.

Ratio of sides \quad 1:1: \sqrt{2}

Right triangle A B C with right angle A. The legs of the triangle have a length of s, and the hypotenuse has a length of s square root of 2. Angles B and C have a measure of 45 degrees.

30\degree-60\degree-90\degree triangle theorem

In a 30\degree-60\degree-90\degree triangle, the length of the hypotenuse is twice the length of the short leg. The length of the long leg is \sqrt{3} times the length of the short leg.

Ratio of sides \quad 1:\sqrt{3}: 2

Right triangle A B C with right angle C. Leg B C has length of s, leg A C has length of s square root of 3, and hypotenuse A B has a length of 2 s. Angle A has a measure of 30 degrees, and angle B has a measure of 60 degrees.

Worked examples

Example 1

Consider the triangle below.

A right triangle with legs of length a and 15, and a hypotenuse of length c. The angle opposite the leg with length a has a measure of 45 degrees.

Find the exact value of a.

Approach

The triangle has a right angle and a 45\degree angle shown. This means that this triangle is a 45\degree-45\degree-90\degree because the sum of interior angles in a triangle is equal to 180 \degree.

Solution

Legs in 45\degree-45\degree-90\degree special right triangles are congruent. We are given that 15 is the length of one leg from the diagram.

a is the other leg of the triangle as a is opposite a 45 \degree angle.

So, a=15.

Find the exact value of c.

Approach

Using the 45\degree-45\degree-90\degree triangle theorem, the hypotenuse is \sqrt{2} times the length of the leg. We want to use this property to find c.

Solution

We know the length of the legs is 15 and c is the hypotenuse as it is opposite the right angle.

So, c=15\sqrt{2}.

Reflection

We could have also used the Pythagorean theorem to solve for c.

Example 2

Consider the triangle below.

A right triangle with legs of length b and 3, and a hypotenuse of length c. The angle opposite the leg with length 3 has a measure of 30 degrees.

Find the exact value of c.

Approach

The triangle has a right angle and a 30 \degree angle. Using the fact that the sum of the interior angles in a triangle is equal to 180 \degree we know that the unlabeled angle is 60 \degree.

This means that the triangle is a 30\degree-60\degree-90\degree special right triangle. This tells us:

The length of the hypotenuse is twice the length of the shorter leg
The side with length 3 is the shortest side as it is opposite the smallest angle.
The side labeled c is the hypotenuse as it is opposite the right angle.

Using the 30\degree-60\degree-90\degree triangle theorem, we can write this equation:

\displaystyle \text{hypotenuse}

\displaystyle =

\displaystyle 2(\text{shorter leg})

We want to use this equation to find c.

Solution

\displaystyle \text{hypotenuse}	\displaystyle =	\displaystyle 2(\text{shorter leg})
\displaystyle c	\displaystyle =	\displaystyle 2(3)	Substitution
\displaystyle c	\displaystyle =	\displaystyle 6	Simplify

So, c=6.

Find the exact value of b.

Approach

From part (a), we know the triangle is a 30\degree-60\degree-90\degree special right triangle. This tells us:

The length of the longer leg is \sqrt{3} times the length of the shorter side.
The longer leg is b since the side is opposite the 60\degree angle.

Using the 30\degree-60\degree-90\degree triangle theorem, we can write this equation:

\displaystyle \text{longer leg}

\displaystyle =

\displaystyle (\text{shorter leg})\sqrt{3}

We want to use this equation to find b.

Solution

\displaystyle \text{longer leg}	\displaystyle =	\displaystyle (\text{shorter leg})\sqrt{3}
\displaystyle b	\displaystyle =	\displaystyle 3\sqrt{3}	Substitution

So, b=3\sqrt{3}.

Outcomes

MA.912.LT.4.8

Construct proofs, including proofs by contradiction.

MA.912.LT.4.10

Judge the validity of arguments and give counterexamples to disprove statements.

MA.912.T.1.2

Solve mathematical and real-world problems involving right triangles using trigonometric ratios and the Pythagorean Theorem.

8.02 Special right triangles

Concept summary

Worked examples

Example 1

Approach

Solution

Approach

Solution

Reflection

Example 2

Approach

Solution

Approach

Solution

Outcomes

MA.912.LT.4.8

MA.912.LT.4.10

MA.912.T.1.2

What is Mathspace

About Mathspace