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.

Year groups: Secondary and Sixth Form, Lower Secondary, Years 9–11
File type: Microsoft Word

Updated Sep 19, 2021

Subject: Geometry, Maths
Topic: Triangle, Equations
Resource types: Worksheets, Worksheets & Printables
England expectations: Key Stage 3 · Geometry and measures

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.

No reviews yet

Downloaded this resource? Tell other teachers how it went.

National Curriculum in England

National Curriculum in England expectations

England expectations

Key Stage 3 · Geometry and measures — England Key Stage 3 Mathematics: use Pythagoras theorem and trigonometric ratios in similar triangles to solve problems involving right-angled triangles.