Lesson 66: Special Right Triangles

New Concept

Two special right triangles, the 45-45-90 and the 30-60-90, have constant side length ratios. All 45-45-90 triangles have side lengths in the ratio of $1:1:\sqrt{2}$, and all 30-60-90 triangles have side lengths in the ratio of $1:\sqrt{3}:2$.

What’s next

This card is just the foundation. Next, you'll apply these ratios in worked examples to find missing side lengths and solve real-world geometry puzzles involving area and distance.

Property

A 45-45-90 triangle is an isosceles right triangle (half a square) whose side lengths are in the ratio $1:1:\sqrt{2}$.

Examples

• If the legs of a 45-45-90 triangle are both 1 unit, the hypotenuse is $1 \cdot \sqrt{2} = \sqrt{2}$ units.
• A baseball diamond is a square with 90-foot sides, so the throw from home to second is the hypotenuse: $90\sqrt{2}$ feet.
• If one leg of a 45-45-90 triangle measures 8 cm, the other leg is also 8 cm and the hypotenuse is $8\sqrt{2}$ cm.

Explanation

Think of this as a perfect square sliced diagonally in half! Because it's an isosceles right triangle, its two legs are always equal. This special relationship means if you know the length of just one leg, you automatically know the other. The hypotenuse is simply the leg length multiplied by the square root of 2.

Property

A 30-60-90 triangle is half of an equilateral triangle, and its side lengths are in the ratio $1:\sqrt{3}:2$.

Examples

• If the shortest side is 1 unit, the other leg is $\sqrt{3}$ units and the hypotenuse is 2 units.
• If the shortest side of a 30-60-90 triangle is 5 inches, the other leg is $5\sqrt{3}$ inches and the hypotenuse is 10 inches.
• If the hypotenuse is 14 meters, the shortest leg is 7 meters and the longer leg is $7\sqrt{3}$ meters.

Explanation

Imagine an equilateral triangle perfectly chopped in half—voilà, a 30-60-90 triangle! The shortest leg is always opposite the tiny 30° angle, and it's exactly half the hypotenuse. The medium leg, across from the 60° angle, is found by multiplying the shortest leg's length by the square root of 3. It’s a predictable family of sides!