Pythagorean Theorem Word Problems Worksheet With Answers
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The Pythagorean Theorem is a fundamental concept in mathematics, especially in geometry. It deals with the relationship between the lengths of the sides of a right triangle. Formulated by the ancient Greek mathematician Pythagoras, the theorem states that 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 theorem has wide applications in various fields, including physics, engineering, architecture, and more. In this article, we'll explore some Pythagorean Theorem word problems, provide worksheets, and discuss answers to help you understand this essential mathematical concept. 📐
Understanding the Pythagorean Theorem
The Pythagorean Theorem can be expressed mathematically as follows:
[ a^2 + b^2 = c^2 ]
Where:
- ( c ) is the length of the hypotenuse.
- ( a ) and ( b ) are the lengths of the other two sides.
To solve problems using this theorem, you must identify the lengths of the sides involved and apply the formula correctly.
Key Terms
- Hypotenuse: The longest side of a right triangle, opposite the right angle.
- Legs: The two shorter sides of a right triangle.
Types of Pythagorean Theorem Word Problems
Pythagorean Theorem word problems can take various forms. Here are some common types:
- Finding the Hypotenuse: Problems where you know the lengths of both legs and need to find the length of the hypotenuse.
- Finding a Leg: Problems where you know the length of the hypotenuse and one leg and need to find the length of the other leg.
- Real-life Applications: Problems involving real-world scenarios such as distance, heights, or diagonal measurements.
Sample Word Problems and Solutions
Let's take a look at a few example problems and their solutions.
Problem 1: Finding the Hypotenuse
Problem: A ladder is leaning against a wall. The bottom of the ladder is 6 feet away from the wall, and the top of the ladder reaches a height of 8 feet up the wall. How long is the ladder?
Solution:
- Identify the sides:
- ( a = 6 ) feet (distance from the wall)
- ( b = 8 ) feet (height of the ladder)
Using the Pythagorean Theorem:
[ a^2 + b^2 = c^2 ] [ 6^2 + 8^2 = c^2 ] [ 36 + 64 = c^2 ] [ 100 = c^2 ] [ c = \sqrt{100} = 10 \text{ feet} ]
The ladder is 10 feet long. 🪜
Problem 2: Finding a Leg
Problem: A triangle has a hypotenuse of 13 meters and one leg measuring 5 meters. What is the length of the other leg?
Solution:
- Identify the sides:
- ( c = 13 ) meters (hypotenuse)
- ( a = 5 ) meters (one leg)
Using the Pythagorean Theorem:
[ a^2 + b^2 = c^2 ] [ 5^2 + b^2 = 13^2 ] [ 25 + b^2 = 169 ] [ b^2 = 169 - 25 ] [ b^2 = 144 ] [ b = \sqrt{144} = 12 \text{ meters} ]
The length of the other leg is 12 meters. 📏
Problem 3: Real-Life Application
Problem: A rectangular park has a length of 30 meters and a width of 40 meters. What is the length of the diagonal across the park?
Solution:
- Identify the sides:
- ( a = 30 ) meters (width)
- ( b = 40 ) meters (length)
Using the Pythagorean Theorem:
[ a^2 + b^2 = c^2 ] [ 30^2 + 40^2 = c^2 ] [ 900 + 1600 = c^2 ] [ 2500 = c^2 ] [ c = \sqrt{2500} = 50 \text{ meters} ]
The diagonal of the park is 50 meters long. 🌳
Practice Worksheet
Here’s a practice worksheet with Pythagorean Theorem word problems for you to solve.
<table> <tr> <th>Problem Number</th> <th>Word Problem</th> </tr> <tr> <td>1</td> <td>A right triangle has legs measuring 9 cm and 12 cm. What is the length of the hypotenuse?</td> </tr> <tr> <td>2</td> <td>A ladder reaches 15 feet high against a wall, with its base 9 feet from the wall. How long is the ladder?</td> </tr> <tr> <td>3</td> <td>The sides of a right triangle are in the ratio 3:4:5. If the shortest side is 6 cm, find the lengths of the other two sides.</td> </tr> <tr> <td>4</td> <td>A rectangular field is 50 meters long and 30 meters wide. What is the length of the diagonal?</td> </tr> </table>
Important Notes
- Always check your calculations: Errors in basic arithmetic can lead to incorrect answers.
- Understand the problem context: Reading the problem carefully helps identify which sides are given and what is being asked.
Conclusion
The Pythagorean Theorem is an invaluable tool for solving geometric problems involving right triangles. Through the examples and practice problems provided, you can develop a solid understanding of this concept. By practicing these word problems, you’ll enhance your mathematical skills and gain confidence in applying the Pythagorean Theorem in various scenarios. Keep practicing, and soon you will be a pro at solving these types of problems! 🌟