Special Right Triangles – Investigation Worksheet

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In the classroom

Teaching tips & learning objectives

Learning objectives

Use the Pythagorean theorem to find side lengths in 45–45–90 triangles
Derive and apply the side-length shortcuts for both special-triangle types
Bisect an equilateral triangle and label the resulting 30–60–90 triangle

Teaching tips

Complete the perfect-squares and radical-simplification warm-up before the triangle problems.
Let students investigate several triangles before revealing the shortcut.
Use the printed summary boxes to check the patterns students discover.

Skills covered

Pythagorean theorem — students calculate unknown sides of isosceles right triangles.
Radical simplification — students simplify square roots used in side-length calculations.
Pattern discovery — students derive special-triangle shortcuts from repeated calculations.

Good to know

Questions teachers ask about this resource

What should students review before starting the investigation?

Review perfect squares, simplifying radicals, and the Pythagorean theorem. The worksheet begins with related warm-ups before students investigate the special-triangle patterns.

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Curriculum

Standards, concepts & topics

Standards

8.G.B.7 — Practices finding unknown right-triangle side lengths with the Pythagorean theorem.

Core concepts

Pythagorean Theorem
Special right triangles
45-45-90 triangle
30-60-90 triangle
Radical simplification

Topics

Special right triangles 45-45-90 triangles 30-60-90 triangles Pythagorean theorem Geometry investigation

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