Triangles: Introduction & Pythagorean Theorem

Page Menu

• Lesson Description
• Learning Objectives
• Topics Covered
• Key Terms
• Lesson Examples
• Practice Problems
• Tips & Common Mistakes
• Summary
• Challenge Problems
• Unit Navigation

Lesson Description

In this lesson, students are introduced to triangles and their basic structure. Special attention is given to right triangles and the Pythagorean Theorem, which allows us to calculate a missing side length when the other two sides are known. This theorem is a foundational tool in geometry and many real-world applications.

Learning Objectives

• Understand the basic parts and types of triangles.
• Identify right triangles and their side relationships.
• Use the Pythagorean Theorem to find missing side lengths.

Topics Covered

• Basic triangle terminology
• Right triangles and side names
• Pythagorean Theorem calculations

Key Terms

• Triangle: A polygon with three sides and three angles.
• Right Triangle: A triangle that contains one 90° angle.
• Hypotenuse: The longest side of a right triangle, opposite the right angle.

Lesson Examples

Example 1: Finding the Hypotenuse

1. A right triangle has legs of length 3 and 4.
2. Apply the formula: a² + b² = c² → 3² + 4² = c².
3. Compute: 9 + 16 = 25, so c = 5.

Example 2: Finding a Missing Leg

1. Hypotenuse is 13 and one leg is 5.
2. Use formula: 5² + b² = 13².
3. Compute: 25 + b² = 169 → b² = 144 → b = 12.

Practice Problems

1. Find the hypotenuse if legs are 6 and 8.
2. Find a missing leg if hypotenuse is 10 and one leg is 6.
3. A right triangle has legs 9 and 12. Find the hypotenuse.
4. If the hypotenuse is 15 and one leg is 9, find the other leg.

Tips & Common Mistakes

• Only use the Pythagorean Theorem with right triangles.
• Make sure the hypotenuse is always the longest side.
• Square numbers carefully to avoid arithmetic mistakes.

Summary

Triangles are key geometric shapes, and right triangles follow a special rule known as the Pythagorean Theorem. By squaring the two legs and adding them, we can find the hypotenuse or solve for a missing side. This concept is essential for future geometry and real-world measurement problems.

Challenge Problems

1. A ladder reaches 24 feet up a wall and sits 7 feet from the wall. How long is the ladder?
2. A right triangle has hypotenuse 25 and one leg 7. Find the other leg.
3. A rectangular field measures 30 m by 40 m. What is the diagonal distance across the field?