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45-45-90 and 30-60-90 Special Right Triangles infographic - Special Right Triangle Ratios

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Geometry

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 2\sqrt{2}. A 30-60-90 triangle comes from splitting an equilateral triangle in half, so its side lengths follow the ratio 1:3:21 : \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:x2x : x : x\sqrt{2}
  • In a 45-45-90 triangle, if each leg = xx, then hypotenuse = x2x\sqrt{2}
  • 30-60-90 side ratio: x:x3:2xx : x\sqrt{3} : 2x
  • In a 30-60-90 triangle, short leg opposite 30 degrees = xx, long leg opposite 60 degrees = x3x\sqrt{3}, hypotenuse = 2x2x
  • If hypotenuse of a 45-45-90 triangle is hh, then each leg = h2=h22\frac{h}{\sqrt{2}} = \frac{h\sqrt{2}}{2}
  • If hypotenuse of a 30-60-90 triangle is hh, then short leg = h2\frac{h}{2} and long leg = h32\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 x2x\sqrt{2} for the hypotenuse of a 30-60-90 triangle, which is wrong because x2x\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. 1 A 45-45-90 triangle has legs of length 8. Find the hypotenuse.
  2. 2 A 30-60-90 triangle has hypotenuse 14. Find the short leg and the long leg.
  3. 3 Explain why a 45-45-90 triangle must have two equal legs and how that determines the hypotenuse formula.