Special Right Triangles – Investigation Worksheet

Apply the Pythagorean Theorem to determine unknown side lengths of isosceles right (45-45-90) triangles.

Discover and articulate the numeric shortcut relating a 45-45-90 triangle's legs to its hypotenuse (multiply by radical 2).

Construct a 30-60-90 triangle by bisecting an equilateral triangle and label its short leg, long leg, and hypotenuse.

Discover and apply the shortcut relating a 30-60-90 triangle's short leg to its hypotenuse and long leg.

Have students complete the perfect-squares list and radical simplification warm-up before the triangle problems, since later shortcuts depend on comfort simplifying radicals.

Let students work through several numbered triangles before revealing the shortcut, since the worksheet is designed as a pattern-discovery investigation rather than a direct-instruction sheet.

Use the closing summary box (with both shortcuts and formulas) as a self-check for students to confirm the pattern they found matches the stated rule.

Pythagorean Theorem application — students solve for unknown sides of isosceles right triangles using the theorem.

Radical simplification — students simplify square-root expressions in preparation for triangle side calculations.

Pattern discovery/shortcut derivation — students notice and state the multiply-by-radical-2 (45-45-90) and double-the-short-leg (30-60-90) shortcuts from repeated practice.

Geometric construction and labelling — students bisect an equilateral triangle and label the resulting short leg, long leg, and hypotenuse.