The Pythagorean Theorem is an important rule in geometry. It helps us understand special right triangles, like the 45-45-90 triangle.

This theorem says that in a right triangle, if you square the length of the longest side (called the hypotenuse), it equals the sum of the squares of the other two sides. In simple terms, you can write it like this:

$$c^2 = a^2 + b^2$$

Here, $c$ is the hypotenuse, and $a$ and $b$ are the lengths of the other two sides.

What is a 45-45-90 Triangle?

A 45-45-90 triangle is a special type of right triangle.

In this triangle:

• Both sides (called legs) are the same length.
• The angles are $45^\circ$, $45^\circ$, and $90^\circ$.

Because of its specific angles, we have a clear relationship between the legs and the hypotenuse.

Key Features of a 45-45-90 Triangle

1. Leg Lengths: If each leg is $x$, you can find the length of the hypotenuse $h$ using the Pythagorean Theorem:

   $$h = x\sqrt{2}$$

2. Ratio of Sides: The ratio of the lengths in a 45-45-90 triangle is:

   • For the legs: $1 : 1$
   • For the hypotenuse: $1 : \sqrt{2}$

How to Solve Problems with 45-45-90 Triangles

If you need to solve a problem involving a 45-45-90 triangle, here are the steps to follow:

1. Check the Triangle: Make sure the triangle you’re looking at is a 45-45-90 triangle. You can do this by checking that two angles are $45^\circ$.

2. Use the Pythagorean Theorem: If you know the lengths of the legs (let's call them $x$), you can use the Pythagorean theorem to check your work:

   $$c^2 = x^2 + x^2 = 2x^2$$

   Then, we take the square root to find:

   $$c = x\sqrt{2}$$

3. Finding Unknown Lengths: If you only know the length of the hypotenuse, you can find the legs. For example, if $c = 10$, you would do:

   $$10 = x\sqrt{2} \implies x = \frac{10}{\sqrt{2}} \approx 7.07$$

Where Do We Use 45-45-90 Triangles?

• In Buildings: Knowing about 45-45-90 triangles is essential when designing buildings that need to be stable and balanced.
• In Everyday Life: Many diagonal supports used in construction are set to make 45-degree angles. Knowing about these triangles helps us in real-world situations.

In short, the Pythagorean Theorem is a strong tool for working with 45-45-90 triangles. It helps us measure lengths and understand how things fit together in various shapes.