Pythagorean Theorem Word Problems: Worksheet Answers Explained
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The Pythagorean Theorem is a fundamental principle in geometry that relates the lengths of the sides of a right triangle. It 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:
[ c^2 = a^2 + b^2 ]
where:
- ( c ) is the length of the hypotenuse,
- ( a ) and ( b ) are the lengths of the other two sides.
In this article, we will dive into common word problems involving the Pythagorean Theorem, providing answers and explanations to help enhance understanding. 🌟
Understanding Word Problems
Word problems can often seem overwhelming, but breaking them down into smaller parts can make them more manageable. Here’s how to approach a Pythagorean theorem word problem:
- Read the problem carefully: Identify what is being asked.
- Sketch the scenario: Visual aids can help you understand relationships between the sides.
- Identify the known and unknown values: Determine which values you already have and which values you need to find.
- Apply the Pythagorean Theorem: Use the theorem to set up your equation and solve for the unknown.
Common Pythagorean Theorem Word Problems
Problem 1: Finding the Length of a Side
Problem Statement: A ladder leans against a wall, reaching a height of 12 feet. The base of the ladder is 5 feet away from the wall. How long is the ladder?
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Identify the knowns:
- Height of the wall (( b )) = 12 feet
- Distance from the wall (( a )) = 5 feet
- Length of the ladder (( c )) = ?
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Apply the Pythagorean Theorem:
[ c^2 = a^2 + b^2 ] [ c^2 = 5^2 + 12^2 ] [ c^2 = 25 + 144 ] [ c^2 = 169 ] [ c = \sqrt{169} = 13 \text{ feet} ]
Answer: The ladder is 13 feet long. 📏
Problem 2: Finding the Height of a Triangle
Problem Statement: A triangular park has a base of 8 meters and a height that forms a right triangle with one of the sides measuring 10 meters. What is the height of the triangle?
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Identify the knowns:
- Base (( a )) = 8 meters
- Side length (( c )) = 10 meters
- Height (( b )) = ?
-
Apply the Pythagorean Theorem:
[ c^2 = a^2 + b^2 ] [ 10^2 = 8^2 + b^2 ] [ 100 = 64 + b^2 ] [ b^2 = 100 - 64 ] [ b^2 = 36 ] [ b = \sqrt{36} = 6 \text{ meters} ]
Answer: The height of the triangle is 6 meters. 🌳
Problem 3: Finding the Distance Between Two Points
Problem Statement: Two points on a coordinate plane are located at (3, 4) and (7, 1). What is the distance between these two points?
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Identify the knowns:
- Point 1: (3, 4)
- Point 2: (7, 1)
Using the distance formula, which is a direct application of the Pythagorean Theorem: [ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]
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Plug in the values: [ d = \sqrt{(7 - 3)^2 + (1 - 4)^2} ] [ d = \sqrt{(4)^2 + (-3)^2} ] [ d = \sqrt{16 + 9} = \sqrt{25} = 5 ]
Answer: The distance between the two points is 5 units. 📍
Practice Problems
To solidify your understanding, try these problems yourself!
- A right triangle has one leg measuring 9 cm and the hypotenuse measuring 15 cm. Find the length of the other leg.
- A rectangular swimming pool is 20 m long and 15 m wide. Find the distance across the pool from one corner to the opposite corner.
Table of Common Values for Quick Reference
<table> <tr> <th>Value</th> <th>Description</th> </tr> <tr> <td>c</td> <td>Hypotenuse</td> </tr> <tr> <td>a, b</td> <td>Legs of the triangle</td> </tr> <tr> <td>√</td> <td>Square root function</td> </tr> <tr> <td>=</td> <td>Equals</td> </tr> </table>
Important Notes
"When solving Pythagorean Theorem problems, always ensure that you identify the right triangle and label the sides correctly. Mislabeling can lead to incorrect calculations."
Understanding the Pythagorean theorem through word problems not only enhances your problem-solving skills but also enables you to see the application of this theorem in real-world situations. Keep practicing and applying these techniques, and soon you will feel confident in tackling any word problem that comes your way! ✨