Help with understanding ratios for special right triangles?

I'm struggling with my geometry class, specifically with the topic of special right triangles (30-60-90 and 45-45-90). Can someone explain their ratios and any useful tips to remember them? Thanks in advance!

Sure, I'd be happy to help with special right triangles! There are two types of special right triangles you've mentioned: 30-60-90 and 45-45-90. Let's break them down.

1. 30-60-90 triangle:

In a 30-60-90 triangle, the angles are 30 degrees, 60 degrees, and 90 degrees. The ratio of the side lengths of this triangle is 1:√3:2. Here's what this means:

- The side opposite the 30-degree angle (shortest side) has length x (we'll call this the "base").

- The side opposite the 60-degree angle has length x√3 (we'll call this the "height").

- The side opposite the 90-degree angle (the hypotenuse) has length 2x.

Remembering this ratio can be helpful for solving problems involving 30-60-90 triangles.

2. 45-45-90 triangle:

In a 45-45-90 triangle, both non-right angles are equal (45 degrees) since its a right isosceles triangle; the other angle, of course, is 90 degrees. The ratio of the side lengths for this triangle is 1:1:√2. Here's how it breaks down:

- The sides opposite the two 45-degree angles (the legs) are equal in length, and we'll call them x.

- The side opposite the 90-degree angle (the hypotenuse) has a length x√2.

Remembering this ratio can be helpful for solving problems involving 45-45-90 triangles.

Here's a helpful tip to remember these ratios: for the 30-60-90 triangle, think "smallest angle, smallest side (x); biggest angle, biggest side (2x), and medium angle, medium side(x√3)." For a 45-45-90 triangle, remember that "45" is half of "90," so the legs are equal, and the hypotenuse has a length equal to one of the legs multiplied by √2.

Practice solving problems with these ratios, and you'll become more comfortable using these properties of special right triangles. Good luck with your geometry class!