45-45-90 and 30-60-90 Special Right Triangles

Special Right Triangle Ratios

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Special right triangles are triangles with fixed angle patterns that always produce the same side ratios. The two most important ones are the 45-45-90 triangle and the 30-60-90 triangle. These triangles matter because they let you find missing side lengths quickly without using a calculator. They appear often in geometry, trigonometry, construction, and coordinate problems.

A 45-45-90 triangle comes from cutting a square along its diagonal, so its legs are equal and its hypotenuse is longer by a factor of $\sqrt{2}$ . A 30-60-90 triangle comes from splitting an equilateral triangle in half, so its side lengths follow the ratio $1 : \sqrt{3} : 2$ . Once you know these patterns, you can scale them to any size triangle with the same angles. This makes many right triangle problems much faster to solve.

Key Facts

45-45-90 side ratio: $x : x : x\sqrt{2}$
In a 45-45-90 triangle, if each leg = $x$ , then hypotenuse = $x\sqrt{2}$
30-60-90 side ratio: $x : x\sqrt{3} : 2x$
In a 30-60-90 triangle, short leg opposite 30 degrees = $x$ , long leg opposite 60 degrees = $x\sqrt{3}$ , hypotenuse = $2x$
If hypotenuse of a 45-45-90 triangle is $h$ , then each leg = $\frac{h}{\sqrt{2}} = \frac{h\sqrt{2}}{2}$
If hypotenuse of a 30-60-90 triangle is $h$ , then short leg = $\frac{h}{2}$ and long leg = $\frac{h\sqrt{3}}{2}$

Vocabulary

Right triangle: A triangle with one angle equal to 90 degrees.
Hypotenuse: The side opposite the 90 degree angle, and it is the longest side of a right triangle.
Leg: Either of the two sides that form the right angle in a right triangle.
Side ratio: A comparison of side lengths that stays the same for all similar triangles of a given type.
Similar triangles: Triangles with the same angle measures and proportional corresponding side lengths.

Common Mistakes to Avoid

Mixing up the long leg and short leg in a 30-60-90 triangle, which is wrong because the short leg is always opposite 30 degrees and the long leg is always opposite 60 degrees.
Using $x\sqrt{2}$ for the hypotenuse of a 30-60-90 triangle, which is wrong because $x\sqrt{2}$ belongs to the 45-45-90 pattern, not the 30-60-90 pattern.
Forgetting that the equal sides in a 45-45-90 triangle are the legs, which is wrong because the hypotenuse is not equal to the legs and must be longer.
Adding side lengths instead of using the special ratio, which is wrong because these triangles depend on multiplicative relationships, not simple addition.

Practice Questions

1 A 45-45-90 triangle has legs of length 8. Find the hypotenuse.
2 A 30-60-90 triangle has hypotenuse 14. Find the short leg and the long leg.
3 Explain why a 45-45-90 triangle must have two equal legs and how that determines the hypotenuse formula.