45-45-90 triangle

A 45-45-90 triangle, also called isosceles right triangle, is a special right triangle in which both legs are congruent and the length of the hypotenuse is the square root of two times the length of a leg.

Hypotenuse = √2 × length of a leg

The legs are congruent

Looking carefully at the figure above, you may have observed the following ratios:

Suppose we start from the smallest angle to the largest angle and from the shortest side to the longest side

The angles of a 45-45-90 triangle are in the ratio 1:1:2

The sides of a 45-45-90 triangle are in the ratio 1:1:√2

45-45-90 triangle proof

Start with an isosceles right triangle ABC like the one shown above.

Let segment AB equal to x and use the Pythagorean theorem to find the length of segment AB.

AB² = AC² + BC²

x² = s² + s²

x² = 2s²

x = √(2s²)

x = (√2)[√(s²)]

x = (√2)(s)

x = s√2

Examples of 45-45-90 triangles

Example #1: 1, 1, √2

Each leg is equal to 1
Hypotenuse: √2

Example #2: 2, 2, 2√2

Each leg is equal to 2
Hypotenuse: 2√2

Example #3: 3, 3, 3√2

Each leg is equal to 3
Hypotenuse: 3√2

Using the length of one side to solve a 45-45-90 triangle

Example #4:

The hypotenuse of a 45-45-90 triangle is 7√2. Find the lengths of the other sides.

Hypotenuse = s√2, s is the length of a leg.

7√2 = s√2

Divide both sides by √2.

7√2 / √2 = s√2 / √2

7 = s

The legs of this 45-45-90 triangle have a length of 7

Example #5:

The legs of a 45-45-90 triangle have a length of 13. Find the length of the hypotenuse.

Hypotenuse = s√2, s is the length of a leg.

The length of the hypotenuse is 13√2

Using the 45-45-90 triangle to solve real-world problems

Example #6:

The distance from first base to second base of a baseball field is usually 90 feet although this distance may vary. Find the distance from home plate to the second base.