45-45-90 Triangle: Finding Missing Sides

In geometry, a 45-45-90 triangle is a special type of right triangle where two of its angles measure 45 degrees each. This unique configuration leads to a specific relationship between the sides of the triangle, making it easier to calculate missing side lengths.

Understanding the Special Ratio

The key to solving 45-45-90 triangles lies in the special ratio of its sides. If one leg of the triangle has a length of ‘x’, the other leg will also have a length of ‘x’, and the hypotenuse (the side opposite the right angle) will have a length of ‘x√2’.

Here’s a visual representation:

Using the Pythagorean Theorem

The Pythagorean Theorem, a fundamental principle in geometry, states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. This theorem can be applied to 45-45-90 triangles to find missing sides.

Let’s say we know the length of one leg is ‘a’. Using the Pythagorean Theorem, we can find the length of the hypotenuse (c):

a² + a² = c²

2a² = c²

c = √(2a²)

c = a√2

Example Problem

Let’s say we have a 45-45-90 triangle where one leg measures 5 units. We can use the special ratio and the Pythagorean Theorem to find the other leg and the hypotenuse:

Step 1: Find the other leg

Since it’s a 45-45-90 triangle, the other leg will also have a length of 5 units.

Step 2: Find the hypotenuse

Using the special ratio, the hypotenuse will be the length of a leg multiplied by √2:

Hypotenuse = 5√2 units

Practice Problems

Try these practice problems to solidify your understanding:

1. If one leg of a 45-45-90 triangle is 8 units long, what is the length of the other leg and the hypotenuse?
2. If the hypotenuse of a 45-45-90 triangle is 10√2 units long, what is the length of each leg?

Conclusion

Understanding the special ratio and the Pythagorean Theorem makes solving 45-45-90 triangles a breeze. Remember, these triangles have two equal angles and a specific relationship between their sides. Practice using these methods to become proficient in working with this unique type of right triangle.