Pythagorean Theorem | MathHelp.com

Introduction

This tutorial will guide you through the Pythagorean Theorem, a fundamental principle in geometry that helps you calculate the lengths of the sides of right triangles. Understanding this theorem is essential for solving various mathematical problems and is widely applicable in fields like architecture, engineering, and physics.

Step 1: Understand the Pythagorean Theorem

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be expressed with the formula:

a² + b² = c²

• a and b are the lengths of the legs of the triangle.
• c is the length of the hypotenuse.

Practical Tip

• Remember that the hypotenuse is always the longest side in a right triangle.

Step 2: Identify the Sides of the Triangle

To apply the Pythagorean Theorem, you first need to identify which sides of the triangle correspond to

a

b

c

• Legs (a and b): The two sides that form the right angle.
• Hypotenuse (c): The side opposite the right angle.

Example

If you have a triangle where one leg measures 3 units and the other leg measures 4 units, you can label them as follows:

• a = 3
• b = 4

Step 3: Apply the Pythagorean Theorem to Find the Missing Side

If you know the lengths of two sides, you can find the length of the third side using the theorem.

Finding the Hypotenuse

1. Use the formula:

   c = √(a² + b²)
   

For our example (a = 3, b = 4)

• Calculate a²: 3² = 9
• Calculate b²: 4² = 16
• Sum the squares: 9 + 16 = 25
• Take the square root: c = √25 = 5

Finding a Leg

If you know the hypotenuse and one leg, rearrange the formula:

a = √(c² - b²)

Example

If c = 5 and b = 4:

1. Calculate c²: 5² = 25
2. Calculate b²: 4² = 16
3. Subtract: 25 - 16 = 9
4. Take the square root: a = √9 = 3

Step 4: Practice with Different Triangles

To solidify your understanding, practice finding the missing side lengths of various right triangles using the Pythagorean Theorem.

Try triangles with different side lengths

• Example 1: a = 6, b = 8
• Example 2: c = 10, b = 6

Common Pitfalls to Avoid

• Forgetting to square the lengths.
• Mixing up the sides (confusing legs with hypotenuse).
• Misapplying the formula when dealing with non-right triangles.

Conclusion

The Pythagorean Theorem is a powerful tool for solving problems involving right triangles. Remember the formula a² + b² = c², identify the sides correctly, and practice with different examples to master this concept. As a next step, consider exploring more complex applications of the theorem in real-world scenarios or delve into trigonometry for a broader understanding of triangles.