Special Right Triangles – Investigation Worksheet

A guided investigation leads students through the Pythagorean-Theorem-based side-length patterns of the 45-45-90 and 30-60-90 special right triangles, ending with a stated shortcut for each.

Grades: Grades 8–10, High School, Middle School
File type: Microsoft Word

Updated Sep 19, 2021

Subject: Geometry, Math
Topic: Triangle, Equations
Resource types: Worksheets, Worksheets & Printables
Ontario expectations: E2.4

From the author

About this resource

This study guide comes with an answer key!

In the classroom

Teaching tips & learning objectives

Learning objectives

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.

Teaching tips

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.

Skills covered

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

Good to know

Questions teachers ask about this resource

The worksheet asks students to find a 'shortcut' before showing it — is the shortcut ever explicitly given if students don't discover it themselves?

Yes — after several practice triangles, the sheet explicitly states each shortcut (e.g., multiply the leg by radical 2 to get the 45-45-90 hypotenuse; double the short leg for the 30-60-90 hypotenuse) in a closing summary box, so students who don't derive it independently still receive it.

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Ontario curriculum

Ontario curriculum expectations

Ontario expectations

E2.4 — Ontario Grade 8 Mathematics E2.4: describe the Pythagorean relationship using various geometric models, and apply the theorem to solve problems involving an unknown side length for a given right triangle.