Pythagorean Theorem Word Problems Worksheet Answers Explained
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The Pythagorean Theorem is one of the most fundamental principles in mathematics, especially in geometry. It relates to the lengths of the sides of a right triangle and is often represented by the equation ( 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. Understanding how to apply this theorem in word problems is essential for students, as these applications help solidify the concept and provide practical problem-solving skills. In this article, we will explore common types of Pythagorean Theorem word problems and provide explanations of their solutions.
Understanding the Pythagorean Theorem
Before diving into word problems, it’s crucial to understand what the Pythagorean Theorem states. It is applicable only to right-angled triangles, which means one angle must measure 90 degrees. The theorem gives a relationship between the lengths of the sides of such triangles:
- Hypotenuse (c): The longest side opposite the right angle.
- Adjacent side (a): One of the two sides forming the right angle.
- Opposite side (b): The other side forming the right angle.
Key Formula: [ a^2 + b^2 = c^2 ]
Common Word Problem Scenarios
Let’s look at some typical scenarios where the Pythagorean Theorem is applied:
1. Finding the Length of the Hypotenuse
Example Problem: 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. How long is the ladder?
Solution: In this scenario, we have:
- ( a = 6 ) (distance from wall)
- ( b = 8 ) (height on the wall)
- ( c = ? ) (length 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 = 10 \text{ feet} ]
2. Finding the Length of a Side
Example Problem: In a right triangle, the hypotenuse measures 13 cm, and one of the legs measures 5 cm. What is the length of the other leg?
Solution: Here, we know:
- ( c = 13 ) (hypotenuse)
- ( a = 5 ) (one leg)
- ( b = ? ) (the other leg)
Using the 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 = 12 \text{ cm} ]
3. Real-World Applications
The Pythagorean Theorem can also be applied in real-world scenarios, such as navigation or construction.
Example Problem: A rectangular field is 40 meters long and 30 meters wide. What is the length of the diagonal that stretches from one corner of the field to the opposite corner?
Solution: Here,
- ( a = 30 ) (width)
- ( b = 40 ) (length)
- ( c = ? ) (diagonal)
Using the formula: [ a^2 + b^2 = c^2 ] [ 30^2 + 40^2 = c^2 ] [ 900 + 1600 = c^2 ] [ 2500 = c^2 ] [ c = 50 \text{ meters} ]
Practice Worksheet Examples
To further help with understanding, here’s a quick practice worksheet that can be utilized for applying the Pythagorean Theorem in various scenarios.
<table> <tr> <th>Problem</th> <th>Solution</th> </tr> <tr> <td>1. A right triangle has legs of 9 and 12. Find the hypotenuse.</td> <td>15</td> </tr> <tr> <td>2. The hypotenuse of a right triangle is 10, and one leg is 6. Find the other leg.</td> <td>8</td> </tr> <tr> <td>3. A triangular garden has one side measuring 7m and another measuring 24m. Find the hypotenuse.</td> <td>25m</td> </tr> </table>
Important Notes
- Visualization is Key: Whenever possible, sketch the problem. Drawing a right triangle can often help visualize the relationships between the sides.
- Check Units: Always ensure that the units for each side of the triangle are the same before applying the Pythagorean Theorem.
- Practice Regularly: The more you practice, the easier these problems will become.
Conclusion
The Pythagorean Theorem is not only a fundamental concept in geometry but also a valuable tool for solving real-world problems. Mastering word problems involving this theorem enhances critical thinking and problem-solving abilities. By practicing various scenarios and applying the theorem consistently, students can develop a solid foundation in mathematics that will benefit them in future mathematical concepts.