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

G.N.Q.A.1

Use units as a way to understand real-world problems.*

G.N.Q.A.1.A

Use appropriate quantities in formulas, converting units as necessary.

G.SRT.C.5

Solve triangles.*

G.SRT.C.5.A

Know and use the Pythagorean Theorem and trigonometric ratios (sine, cosine, tangent, and their inverses) to solve right triangles in a real-world context.

G.MP1

Make sense of problems and persevere in solving them.

G.MP3

Construct viable arguments and critique the reasoning of others.

G.MP4

Model with mathematics.

G.MP5

Use appropriate tools strategically.

G.MP6

Attend to precision.

G.MP7

Look for and make use of structure.

G.MP8

Look for and express regularity in repeated reasoning.

8.02 Special right triangles

Concept summary

Worked examples

Example 1

Approach

Solution

Approach

Solution

Reflection

Example 2

Approach

Solution

Approach

Solution

Outcomes

G.N.Q.A.1

G.N.Q.A.1.A

G.SRT.C.5

G.SRT.C.5.A

G.MP1

G.MP3

G.MP4

G.MP5

G.MP6

G.MP7

G.MP8

What is Mathspace

About Mathspace