Special Right Triangles - Interactive Activity

Definitions and Main Ideas

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

1. The names 45-45-90 and 30-60-90 describe the angle measures of the triangles, and the side patterns follow from those angles.
2. 45-45-90 pattern: leg, leg, leg√2. To go from the hypotenuse back to a leg, divide by √2.
3. 30-60-90 pattern: short leg x, long leg x√3, hypotenuse 2x — all three sides come from the one value x.
4. Given the long leg of a 30-60-90, recover the short leg by dividing by √3, then double it for the hypotenuse.
5. Keep answers in simplest radical form unless a rounded decimal is requested.

The Patterns as Formulas

45-45-90: Hypotenuse = Leg · √2

30-60-90: Hypotenuse = 2 · Short Leg, and Long Leg = Short Leg · √3

Special Triangle Explorer

Pick a kind of triangle — any right triangle, 45-45-90, or 30-60-90. Drag the orange handle to resize the triangle and watch the three side lengths follow the pattern for the kind you picked.

Find The Missing Side

One side is given and a second side is highlighted. Use the pattern for the kind of triangle shown to find the highlighted side's measure, in simplest radical form. Watch which side is given — some problems work the pattern backward.

Build the Sides

Work each problem one step at a time: choose the equation that applies the pattern, then isolate and simplify, keeping every answer in simplest radical form. The diagram shows only what you have already found — the highlighted side keeps its starting label until its step is answered.

Your Turn

Answer each question. A first miss gives a hint and another try, and a second miss shows the full explanation. Your score counts first-try answers only.