Absolutely! Let's explore the cool world of 45-45-90 triangles and how they relate to the Pythagorean Theorem! 🎉

What is a 45-45-90 Triangle?

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

It has three angles:

• Two angles are 45 degrees
• One angle is 90 degrees.

What's neat about these triangles is that they are isosceles, meaning the two shorter sides (or legs) are the same length!

If we call the length of each leg "x", then the longest side (called the hypotenuse) will be "x times the square root of 2" (written as (x\sqrt{2})).

Pythagorean Theorem Connection

The Pythagorean Theorem is an important rule that works for all right triangles. It says:

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

In this formula:

• (a) and (b) are the lengths of the two legs.
• (c) is the length of the hypotenuse.

For our 45-45-90 triangle:

• Both legs are (x): $a = b = x$
• The hypotenuse is: $c = x\sqrt{2}$

Putting it Together

Now let's use the Pythagorean Theorem with our triangle:

1. Substitute (x) for (a) and (b):

$$x^2 + x^2 = (x\sqrt{2})^2$$

2. This gives us:

$$2x^2 = 2x^2$$

This shows that our triangle really follows the Pythagorean Theorem! 🌟

Understanding 45-45-90 triangles makes math a bit easier and helps us learn important ideas in trigonometry! Keep your curiosity alive as you explore these wonderful triangles! 🎈