Lesson 8.1: Pythagorean Theorem & Special Triangles Calculations

Learning Objectives

• Apply the Pythagorean Theorem to find missing sides.
• Identify and use 45-45-90 and 30-60-90 triangle ratios.
• Simplify radical expressions accurately.

Topics Covered

• Pythagorean Theorem: \( a^2 + b^2 = c^2 \)
• Identifying the hypotenuse and legs
• 45-45-90 triangle ratio: \( x, x, x\sqrt{2} \)
• 30-60-90 triangle ratio: \( x, x\sqrt{3}, 2x \)
• Radical simplification

Key Terms

• Pythagorean Theorem: Relationship between sides of a right triangle.
• Legs: The two sides forming the right angle.
• Hypotenuse: The side opposite the right angle.
• Special Right Triangle: A right triangle with predictable side ratios.

Lesson Examples

Example 1: Using the Pythagorean Theorem

Given legs 6 and 8, find the hypotenuse.

\[ c^2 = 6^2 + 8^2 = 36 + 64 = 100 \]

\[ c = 10 \]

Example 2: 45-45-90 Triangle

If a leg is 5, the hypotenuse is:

\[ 5\sqrt{2} \]

Example 3: 30-60-90 Triangle

If the shortest leg is 4, then:

Longer leg = \(4\sqrt{3}\)

Hypotenuse = \(8\)

Practice Problems

1. Find the hypotenuse if legs are 9 and 12.
2. In a 45-45-90 triangle, if one leg is 7, find the hypotenuse.
3. In a 30-60-90 triangle, if the hypotenuse is 10, find the shorter leg.

Tips & Common Mistakes

• Always verify which side is the hypotenuse before applying the formula.
• Do not round prematurely unless instructed.
• Simplify radicals fully.

Summary

The Pythagorean Theorem and special right triangle ratios provide efficient tools for solving right triangle problems. Mastery of these relationships reduces computational complexity and improves problem-solving speed.

Challenge Problems

1. A ladder 13 feet long leans against a wall. The base is 5 feet from the wall. How high does it reach?
2. A square has diagonal length 12. Find the side length.
3. A 30-60-90 triangle has longer leg 9√3. Find the hypotenuse.