pythagorean theorem word problems worksheet

Neter

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23-11-2024

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The Pythagorean Theorem is a fundamental concept in geometry, with applications extending far beyond the classroom. This worksheet will help you solidify your understanding by tackling a variety of word problems. Remember, the theorem states: a² + b² = c², where 'a' and 'b' are the legs of a right triangle, and 'c' is the hypotenuse (the side opposite the right angle).

Understanding the Fundamentals: Before We Begin

Before diving into the word problems, let's refresh our understanding of the Pythagorean Theorem and its components:

• Right Triangle: A triangle with one 90-degree (right) angle.
• Legs (a and b): The two sides that form the right angle.
• Hypotenuse (c): The side opposite the right angle; always the longest side.

The Pythagorean Theorem allows us to calculate the length of an unknown side if we know the lengths of the other two sides. Let's apply this to real-world scenarios.

Pythagorean Theorem Word Problems

Here are some word problems to test your skills. Remember to draw a diagram to visualize each problem!

Problem 1: The Ladder

A 10-foot ladder is leaning against a wall. The base of the ladder is 6 feet away from the wall. How high up the wall does the ladder reach?

Solution:

1. Draw a diagram: Sketch a right triangle with the ladder as the hypotenuse (10 feet), the distance from the wall to the base of the ladder as one leg (6 feet), and the height the ladder reaches up the wall as the other leg (unknown).

2. Apply the Pythagorean Theorem: Let 'x' represent the height. Then, 6² + x² = 10².

3. Solve for x: 36 + x² = 100 => x² = 64 => x = 8 feet.

Problem 2: The Diagonal of a Rectangle

A rectangular garden is 12 meters long and 5 meters wide. What is the length of the diagonal path across the garden?

Solution:

1. Draw a diagram: Visualize the rectangle with its diagonal forming the hypotenuse of a right triangle. The length and width are the legs.

2. Apply the Pythagorean Theorem: 12² + 5² = c²

3. Solve for c: 144 + 25 = c² => c² = 169 => c = 13 meters.

Problem 3: The Television Screen

A television screen is advertised as a 50-inch screen. This refers to the diagonal measurement. If the screen’s width is 40 inches, what is its height?

Solution:

1. Draw a diagram: The diagonal is the hypotenuse, the width is one leg, and the height is the other leg.

2. Apply the Pythagorean Theorem: 40² + h² = 50² (where h is the height)

3. Solve for h: 1600 + h² = 2500 => h² = 900 => h = 30 inches.

Problem 4: The Baseball Diamond

A baseball diamond is a square with sides of 90 feet. What is the distance from home plate to second base (the diagonal)?

Solution:

1. Draw a diagram: The diamond is a square, so the diagonal forms the hypotenuse of a right isosceles triangle.

2. Apply the Pythagorean Theorem: 90² + 90² = c²

3. Solve for c: 8100 + 8100 = c² => c² = 16200 => c ≈ 127.3 feet.

Problem 5: The Airplane's Flight Path

An airplane flies 200 miles due north, then 150 miles due east. How far is the airplane from its starting point?

Solution:

1. Draw a diagram: The north and east directions form the legs of a right triangle. The distance from the starting point is the hypotenuse.

2. Apply the Pythagorean Theorem: 200² + 150² = c²

3. Solve for c: 40000 + 22500 = c² => c² = 62500 => c ≈ 250 miles.

Practice Makes Perfect!

These are just a few examples. Creating more word problems using different real-world scenarios will help you grasp the concept of the Pythagorean Theorem even better. Remember to always draw a diagram and carefully label the sides before applying the theorem. Good luck!