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Last modified on August 3rd, 2023

A 45-45-90 triangle is a special right triangle that has two 45-degree angles and one 90-degree angle.

It is also sometimes called an isosceles right triangle since it has two equal sides and two equal angles. It is a special triangle because its side lengths are always in the ratio of 1:1: √2 (x: x: x√2 for side1: side2: hypotenuse).

The 45-45-90 triangle is half of a square. This is because a square has all four angles measuring 90° each. When a square is cut diagonally, one angle remains 90° and the other two 90° angles bisected and become 45° each. The diagonal of a square becomes hypotenuse of a right triangle and the other two sides become the base and the height of the 45-45-90 triangle.

Let side 1 and side 2 of an isosceles-right be x.

We can calculate the hypotenuse of a 45-45-90 right triangle applying the Pythagoras formula a^{2} + b^{2} = c^{2}, where a = side 1, b = side 2, and c = hypotenuse.

Hypotenuse = x^{2} + x^{2} = 2x^{2}

x + x = x√2

- Has two equal side lengths and two equal angles and thus the only possible right-triangle, which is isosceles
- The angles are in the ratio 1: 1: 2
- The sides are in the ratio 1: 1: √2 (x: x: x√2)
- Has one line of symmetry
- Has no rotational symmetry

Thus, the most important rule of a 45-45-90 triangle is that it has one right angle and two other angles equal to 45°. This means, it has two sides of equal length and an unequal side called the base.

Given the length of one side of a 45-45-90 triangle, we can easily calculate the other missing side lengths and also the area and perimeter of the triangle without using the Pythagorean Theorem or trigonometric functions.

Solving a 45-45-90 triangle can have two possibilities:

**Possibility 1**: To calculate hypotenuse when the length of one side is given, multiply the given length by √2**Possibility 2**: To calculate the side lengths given the length of the hypotenuse, divide the hypotenuse by √2

Thus, all problems regarding 45-45-90 triangle can be solved using the 1:1: √2 ratio method.

The formulas of a 45-45-90 triangle when the lengths of its equal sides are ‘x’ units each, are given below:

Let us solve some examples with both the possibilities to understand the concepts better.

**The hypotenuse of a 45-45-90 triangle is 8√2 m. Calculate the length of its base and height.**

Solution:

As we know,

Ratios of a 45-45-90 triangle side lengths = x: x: x√2, here x√2 = hypotenuse = 8√2 mm

Thus,

x√2 = 8√2 m

Squaring both sides we get,

⇒ (x√2)^{2} = (8√2)^{2} m

⇒ 2x^{2} = 64 x 2

⇒ x^{2} = 64

⇒ x = 8 m

Hence, the base and height of the given 45-45-90 triangle is 8 m each.

**Solve the side lengths of a right triangle whose one angle is 45° and the hypotenuse is 6√2 cm.**

Solution:

Given, one angle measures 45°, the given triangle is thus a 45-45-90 triangle.

Hence, we will use x: x: x√2 ratio of side lengths, here x√2 = hypotenuse = 6√2 cm

Dividing both sides of the equation by √2, we get

⇒ 6√2/√2 = x√2/√2

⇒ 6 = x

⇒ x = 6

Hence, the length of each side of the given 45-45-90 triangle is 6 cm.

**The shorter side of an isosceles right triangle is 5√2 mm. What is the diagonal of the triangle?**

Solution:

Since an isosceles right triangle is the same as a 45-45-90 triangle, here we will use x: x: x√2 ratios of side lengths to calculate the length of the hypotenuse, which is the diagonal.

Here, x = 5√2

Thus, for finding the hypotenuse,

x√2 = (5√2)√2

x√2 = 5 mm

Hence, the hypotenuse of the given 45-45-90 triangle is 5 mm.

**The angle of elevation of the top of a office building from the ground is 8 cm from the base of the building is 45°. Find the elevation length of the building.**

Solution:

Since, angle of elevation makes a 45-45-90 triangle with the ground,

Its side lengths = x: x: x√2, here x = 8 cm

Since,

Elevation length = hypotenuse

Substituting the value of x, we get,

x√2 = 8√2 cm

Hence, the elevation length of the top of the office building from the ground is 8√2 cm.

Last modified on August 3rd, 2023