Master

45-45-90 Triangles

Master 45-45-90 triangles with interactive lessons and practice problems! Designed for students like you!

Learn

Get the basics

Practice

Apply your skills

Master

Achieve excellence

45-45-90 Triangles

Study Guide

Key Definition

A 45-45-90 triangle is a special type of isosceles right triangle where the two legs are congruent and the hypotenuse is $\sqrt{2}$ times the length of each leg.

Important Notes

The two legs of a 45-45-90 triangle are of equal length.
The hypotenuse is $\sqrt{2}$ times the length of a leg.
The angles in a 45-45-90 triangle are 45°, 45°, and 90°.

Mathematical Notation

$\sqrt{2}$ represents the square root of 2

$x^2$ represents a number (x) raised to the power of 2

The hypotenuse is the leg length multiplied by √2.

Why It Works

The properties of a 45-45-90 triangle come from the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

Remember

A 45-45-90 triangle has sides in the ratio $1 : 1 : \sqrt{2}$.

Quick Reference

Side Ratio:$1 : 1 : \sqrt{2}$

Understanding 45-45-90 Triangles

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

A 45-45-90 triangle is a right triangle with the two non-right angles each measuring 45 degrees.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the ratio of sides in a 45-45-90 triangle?

Real-World Problem

Question Exercise

Intermediate

Architectural Scenario

An architect is designing a symmetrical roof with equal slopes on either side. The total base is 20 feet wide, which splits evenly into two 10-foot sections. What is the length of each sloping rafter (the rafter length)?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

What is the area of a 45-45-90 triangle if the length of one leg is 5 units?

Challenge Quiz

Single Choice Quiz

Advanced

What is the length of the hypotenuse in a 45-45-90 triangle if one leg measures 7 units?

Recap

Watch & Learn

Review key concepts and takeaways