Pythagorean Theorem Word Problems Worksheet For Practice
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The Pythagorean Theorem is a fundamental principle in geometry that establishes a relationship between the sides of a right triangle. It can be articulated with the formula: (a^2 + b^2 = c^2), where (c) is the length of the hypotenuse, and (a) and (b) are the lengths of the other two sides. This theorem not only aids in theoretical math but also finds practical applications in various fields such as architecture, construction, navigation, and more.
To gain a deeper understanding of this theorem, especially for students, practicing Pythagorean Theorem word problems can be highly beneficial. In this article, we will discuss how to approach these word problems effectively and provide examples for practice.
Understanding the Pythagorean Theorem
What is the Pythagorean Theorem?
The Pythagorean Theorem states that in a right triangle:
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The square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a) and (b).
Mathematically, it is expressed as:
[ a^2 + b^2 = c^2 ]
Why is it Important?
The theorem is crucial in many real-life scenarios:
- Construction and Architecture: Ensuring walls are straight and structures are stable.
- Navigation: Helping in determining the shortest path between two points.
- Computer Graphics: Calculating distances and rendering objects accurately.
Types of Word Problems
Word problems can take various forms, and here are a few common types related to the Pythagorean Theorem:
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Finding the Length of the Hypotenuse:
- Given two sides, find the length of the hypotenuse.
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Finding the Length of One Side:
- Given the length of the hypotenuse and one side, find the other side.
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Real-Life Applications:
- Problems that describe scenarios using the theorem to find distances, heights, etc.
Sample Word Problems for Practice
Let’s look at some examples of word problems to practice applying the Pythagorean Theorem.
Problem 1: Finding the Hypotenuse
A ladder is leaning against a wall. The foot of the ladder is 6 feet away from the wall, and the top of the ladder reaches a height of 8 feet on the wall. What is the length of the ladder?
Solution:
- Let (a = 6) feet (distance from the wall) and (b = 8) feet (height on the wall).
- We need to find (c) (the length of the ladder).
Using the Pythagorean Theorem:
[ 6^2 + 8^2 = c^2 ] [ 36 + 64 = c^2 ] [ 100 = c^2 ] [ c = 10 \text{ feet} ]
The length of the ladder is 10 feet. 🪜
Problem 2: Finding a Missing Side
A rectangular park has one side measuring 10 meters and a diagonal path that measures 12 meters. What is the length of the other side of the park?
Solution:
- Let (a = 10) meters (one side), (c = 12) meters (diagonal).
- We need to find (b) (the other side).
Using the Pythagorean Theorem:
[ 10^2 + b^2 = 12^2 ] [ 100 + b^2 = 144 ] [ b^2 = 144 - 100 ] [ b^2 = 44 ] [ b = \sqrt{44} \approx 6.63 \text{ meters} ]
The length of the other side is approximately 6.63 meters. 🌳
Problem 3: Real-Life Application
A television screen measures 32 inches diagonally. If the width of the screen is 28 inches, how tall is the screen?
Solution:
- Let (a = 28) inches (width), (c = 32) inches (diagonal).
- We need to find (b) (height).
Using the Pythagorean Theorem:
[ 28^2 + b^2 = 32^2 ] [ 784 + b^2 = 1024 ] [ b^2 = 1024 - 784 ] [ b^2 = 240 ] [ b = \sqrt{240} \approx 15.49 \text{ inches} ]
The height of the screen is approximately 15.49 inches. 📺
Practice Problems
Here are some problems for you to practice:
<table> <tr> <th>Problem Number</th> <th>Word Problem</th> </tr> <tr> <td>1</td> <td>A right triangle has one leg measuring 9 cm. If the hypotenuse measures 15 cm, what is the length of the other leg?</td> </tr> <tr> <td>2</td> <td>The base of a tree is 3 meters away from a wall. If the height of the tree is 4 meters, how long is the rope that stretches from the top of the tree to the base of the wall?</td> </tr> <tr> <td>3</td> <td>Two buildings are 20 meters apart. One building is 15 meters tall. What is the distance between the top of the building and the base of the other building?</td> </tr> </table>
Important Note: "Ensure to show your workings for each problem using the Pythagorean theorem."
Conclusion
By practicing these Pythagorean Theorem word problems, students can solidify their understanding and application of this fundamental theorem. Whether in academics or real-world scenarios, the ability to utilize this theorem efficiently will enhance problem-solving skills significantly. Happy practicing! 📚✨