Pythagorean Theorem Worksheet With Answer Key For Students
Table of Contents :
The Pythagorean Theorem is a cornerstone of geometry, bridging the realms of algebra, measurement, and real-world problem solving. It's a fundamental concept that every student should grasp, especially when learning about right triangles. A well-structured worksheet along with an answer key can enhance students' understanding and retention of this important theorem.
Understanding 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 mathematically as:
c² = a² + b²
Where:
- c = length of the hypotenuse
- a and b = lengths of the other two sides
Application of the Pythagorean Theorem
Understanding how to apply the theorem is crucial. It's not just about knowing the formula; it’s about how to use it in various scenarios, from determining the lengths of sides to solving real-world problems.
Pythagorean Theorem Worksheet
To effectively practice the Pythagorean Theorem, here is a sample worksheet designed for students:
Part 1: Basic Problems
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A right triangle has one side measuring 3 cm and another measuring 4 cm. What is the length of the hypotenuse?
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The legs of a right triangle are 5 m and 12 m long. Find the hypotenuse.
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If the hypotenuse of a right triangle is 13 inches and one leg is 5 inches long, what is the length of the other leg?
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A right triangle has a hypotenuse of 10 feet and one leg measuring 6 feet. What is the length of the other leg?
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The legs of a right triangle measure 8 km and 15 km. Calculate the hypotenuse.
Part 2: Word Problems
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A ladder is leaning against a wall. The bottom of the ladder is 6 feet from the wall, and the ladder reaches a height of 8 feet on the wall. How long is the ladder?
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A rectangular park has a diagonal that measures 50 m. One side of the park is 30 m. What is the length of the other side?
Table of Problems
<table> <tr> <th>Problem Number</th> <th>Type</th> <th>Details</th> </tr> <tr> <td>1</td> <td>Basic</td> <td>3 cm, 4 cm</td> </tr> <tr> <td>2</td> <td>Basic</td> <td>5 m, 12 m</td> </tr> <tr> <td>3</td> <td>Basic</td> <td>Hypotenuse: 13 in, one leg: 5 in</td> </tr> <tr> <td>4</td> <td>Basic</td> <td>Hypotenuse: 10 ft, one leg: 6 ft</td> </tr> <tr> <td>5</td> <td>Basic</td> <td>8 km, 15 km</td> </tr> <tr> <td>6</td> <td>Word Problem</td> <td>Ladder: 6 ft from wall, height 8 ft</td> </tr> <tr> <td>7</td> <td>Word Problem</td> <td>Diagonal: 50 m, one side: 30 m</td> </tr> </table>
Tips for Solving Pythagorean Theorem Problems
- Identify the Right Triangle: Ensure the triangle is a right triangle before applying the theorem.
- Label the Sides: Clearly label the hypotenuse and the two legs to avoid confusion.
- Square the Lengths: When using the formula, remember to square the lengths of the legs before adding them.
- Take the Square Root: For finding the hypotenuse, take the square root of the sum of the squares of the legs.
Answer Key for the Pythagorean Theorem Worksheet
Answers to Part 1: Basic Problems
- Answer: 5 cm
- Explanation: c² = 3² + 4² → c² = 9 + 16 → c² = 25 → c = √25 = 5 cm
- Answer: 13 m
- Explanation: c² = 5² + 12² → c² = 25 + 144 → c² = 169 → c = √169 = 13 m
- Answer: 12 inches
- Explanation: 13² = 5² + b² → 169 = 25 + b² → b² = 144 → b = √144 = 12 inches
- Answer: 8 feet
- Explanation: 10² = 6² + b² → 100 = 36 + b² → b² = 64 → b = √64 = 8 feet
- Answer: 17 km
- Explanation: c² = 8² + 15² → c² = 64 + 225 → c² = 289 → c = √289 = 17 km
Answers to Part 2: Word Problems
- Answer: 10 feet
- Explanation: c² = 6² + 8² → c² = 36 + 64 → c² = 100 → c = √100 = 10 feet
- Answer: 40 m
- Explanation: 50² = 30² + b² → 2500 = 900 + b² → b² = 1600 → b = √1600 = 40 m
Conclusion
Incorporating worksheets focused on the Pythagorean Theorem into your study routine can significantly boost your confidence and skills in geometry. These exercises provide essential practice for mastering the theorem, which is applicable in various fields, including physics, engineering, and architecture. By taking the time to work through these problems, students will not only enhance their understanding of geometry but also build critical problem-solving skills that will benefit them in their academic pursuits and beyond. Happy learning! 📚✨