Special Right Triangles
Overview
Two right triangles turn up so often that their side ratios are worth knowing cold. In a 45-45-90 triangle the legs are equal and the hypotenuse is a leg times √2; in a 30-60-90 triangle the hypotenuse is twice the short leg and the long leg is the short leg times √3. This lesson uses those patterns in both directions — building up from a leg, or working back from the hypotenuse or long leg — keeping answers in simplest radical form.
Vocabulary
- 45-45-90 Triangle
- A right triangle whose acute angles each measure 45°. It is a right isosceles triangle: the legs are congruent, and the hypotenuse equals a leg times √2.
- 30-60-90 Triangle
- A right triangle whose acute angles measure 30° and 60°. The hypotenuse is twice the length of the short leg, and the long leg is the short leg times √3.
- Short Leg
- In a 30-60-90 triangle, the leg opposite the 30° angle. It is the shortest side of the triangle.
- Long Leg
- In a 30-60-90 triangle, the leg opposite the 60° angle. Its length is the short leg times √3.
- Simplest Radical Form
- A radical expression simplified so that no perfect-square factor remains under the radical sign and no radical remains in a denominator.
Main Ideas
- The names 45-45-90 and 30-60-90 describe the angle measures of the triangles, and the side patterns follow from those angles.
- 45-45-90 pattern: leg, leg, leg√2. To go from the hypotenuse back to a leg, divide by √2.
- 30-60-90 pattern: short leg x, long leg x√3, hypotenuse 2x — all three sides come from the one value x.
- Given the long leg of a 30-60-90, recover the short leg by dividing by √3, then double it for the hypotenuse.
- Keep answers in simplest radical form unless a rounded decimal is requested.
Lesson Notes
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Lesson Video
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Practice Problems
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Download the practice worksheet for this lesson, work the problems, then check your work with the solutions.