45-45-90 Triangle Guide

Introduction

• Focus on the 45-45-90 triangle.
• Important for solving other triangles and useful for SAT/HT exams.

Understanding the 45-45-90 Triangle

• It's a right triangle (one 90-degree angle).
• Two sides (legs) are equal.
• Hypotenuse is the leg length times the square root of two.

Calculating Sides

• Given leg length:
  • Hypotenuse = Leg length x √2.
• Given hypotenuse:
  • Leg length = Hypotenuse / √2.

Example Problems

Example 1

• Given one leg = 8, Find missing sides.
  • Other leg = 8 (45-45-90 properties).
  • Hypotenuse = 8√2.

Example 2

• Given other leg = 12.
  • Hypotenuse = 12√2.

Example 3

• Given one leg = 5√2.
  • Hypotenuse = 10.

Example 4

• Given hypotenuse = 13√2.
  • Legs = 13.

Rationalizing the Denominator

• Given hypotenuse = 10 or 15.
  • Divide by √2 and rationalize the denominator.
  • Example: 10/√2 → Multiply numerator and denominator by √2 → Results in 5√2.
  • Example: 15/√2 → Results in 15√2/2.

Practice Problem

• Problem Statement: Segment AC is 16 units, B is center of the circle.
• Identify Triangle Type: 45-45-90 (isosceles right triangle).
• Reasoning: AB and BC (radii) are equal.

Solution Steps

• Hypotenuse = 16.
• Find legs: 16/√2 → Rationalize → Legs = 8√2.
• Calculate Area:
  • Area = 1/2 x base x height.
  • Base = 8√2, Height = 8√2.
  • Area = 64 square units.

Summary

• Key Rule:
  • Leg to hypotenuse: Multiply by √2.
  • Hypotenuse to leg: Divide by √2.
• Useful for quick calculations in exams like SAT.