Pythagorean Theorem Word Problems: Answer Key & Solutions
Table of Contents :
The Pythagorean theorem is a fundamental concept in mathematics that helps us understand the relationship between the sides of a right triangle. This 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. In mathematical terms, it can be expressed as:
[ 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.
To illustrate how this theorem can be applied, we will dive into a series of word problems followed by their answers and solutions. This approach will provide clarity on how to utilize the Pythagorean theorem effectively in real-world scenarios.
Understanding the Pythagorean Theorem
The Pythagorean theorem is not just limited to textbook problems; it can be applied in various real-life situations. Here are a few key areas where this theorem plays a crucial role:
- Architecture and Construction 🏗️
- Navigation and Surveying 🧭
- Physics and Engineering ⚙️
Understanding how to solve these problems is vital for students and professionals alike.
Sample Word Problems
Let's explore some word problems involving the Pythagorean theorem, then provide the answer key and solutions for each.
Problem 1: Finding the Length of a Ladder
A ladder is leaning against a wall. The base of the ladder is 4 feet away from the wall, and the top of the ladder reaches a height of 3 feet on the wall. How long is the ladder?
Problem 2: Triangle Side Lengths
In a right triangle, one leg measures 6 cm, and the other leg measures 8 cm. What is the length of the hypotenuse?
Problem 3: Diagonal Distance
A rectangular park has a length of 50 meters and a width of 30 meters. What is the distance between two opposite corners of the park?
Problem 4: Ladder Positioning
A fireman needs to place a 20-foot ladder against a building. If the base of the ladder is 9 feet from the building, how high up the building does the ladder reach?
Answer Key
Below is a summarized answer key for the problems listed above.
<table> <tr> <th>Problem</th> <th>Answer</th> </tr> <tr> <td>Problem 1</td> <td>5 feet</td> </tr> <tr> <td>Problem 2</td> <td>10 cm</td> </tr> <tr> <td>Problem 3</td> <td>58.31 meters</td> </tr> <tr> <td>Problem 4</td> <td>√(20² - 9²) = 17 feet</td> </tr> </table>
Solutions to the Problems
Solution 1: Finding the Length of a Ladder
For Problem 1, we have:
- Base (a) = 4 feet
- Height (b) = 3 feet
Using the Pythagorean theorem: [ c^2 = a^2 + b^2 ] [ c^2 = 4^2 + 3^2 ] [ c^2 = 16 + 9 ] [ c^2 = 25 ] [ c = \sqrt{25} = 5 ]
Therefore, the length of the ladder is 5 feet.
Solution 2: Triangle Side Lengths
For Problem 2, given:
- Leg 1 (a) = 6 cm
- Leg 2 (b) = 8 cm
We apply the theorem: [ c^2 = a^2 + b^2 ] [ c^2 = 6^2 + 8^2 ] [ c^2 = 36 + 64 ] [ c^2 = 100 ] [ c = \sqrt{100} = 10 ]
Hence, the length of the hypotenuse is 10 cm.
Solution 3: Diagonal Distance
For Problem 3, we have:
- Length (a) = 50 meters
- Width (b) = 30 meters
Using the theorem: [ c^2 = a^2 + b^2 ] [ c^2 = 50^2 + 30^2 ] [ c^2 = 2500 + 900 ] [ c^2 = 3400 ] [ c = \sqrt{3400} \approx 58.31 ]
Thus, the distance between the corners is approximately 58.31 meters.
Solution 4: Ladder Positioning
In Problem 4, we know:
- Length of ladder (c) = 20 feet
- Distance from building (a) = 9 feet
We need to find b: [ b^2 = c^2 - a^2 ] [ b^2 = 20^2 - 9^2 ] [ b^2 = 400 - 81 ] [ b^2 = 319 ] [ b = \sqrt{319} \approx 17.88 ]
So, the height up the building is approximately 17 feet.
Important Notes
-
Always check if you have a right triangle: The Pythagorean theorem only applies to right triangles. Ensure the triangle you are working with meets this criterion.
-
Units Matter: When calculating, ensure you are consistent with your units (meters, feet, etc.).
-
Double-check your calculations: Mistakes can happen, so it's always good to verify your answers.
By practicing these types of problems, you can strengthen your understanding of the Pythagorean theorem and improve your problem-solving skills in geometry. Whether you're a student studying for an exam or a professional applying these principles in real life, mastering this theorem is invaluable.