# 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. The distance x from home plate to the second base is the length of the hypotenuse of a 45-45-90 triangle as shown in the figure below. Hypotenuse = s√2, s is the length of a leg.

Since the distance from first base to second base is 90 feet, s is the length of the leg of the 45-45-90 degree triangle.

Hypotenuse = s√2

Hypotenuse = 90√2

Hypotenuse = 127.278

The distance from home plate to the second base is 127.278 feet.

Example #7:

A gardener wants to make a square garden whose diagonal is equal to 45√2 feet . What is the perimeter of the garden?

First, you need to find the length of each side of the square using the formula below.

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

45√2 = s√2

Divide both sides by √2 to find the s

45√2 / √2 = s√2 / √2

s = 45 feet

The length of each side is 45 feet.

The perimeter of the square is 45 + 45 + 45 + 45 = 90 + 90 = 180

The perimeter of the square is 180 feet.

## Area of a 45-45-90 triangle

Finding the area of a 45-45-90 triangle is very straightforward. If s is the length of a leg, then, the area of a 45-45-90 triangle is s2

Example #8:

Find the area of a 45-45-90 triangle if the length of a leg is 25√3

Area  = s2 = (25√3)2

Area  = (25)2(√3)2

Area  = 625(√3)(√3)

Area  = 625(√9)

Area  = 625(3)

Area = 1875

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