Pythagorean Theorem Word Problems Worksheets & Answers
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The Pythagorean Theorem is one of the fundamental principles in mathematics, particularly in geometry. It describes a special relationship between the sides of a right triangle, which is expressed through the formula: a² + b² = c², where c represents the length of the hypotenuse and a and b represent the lengths of the other two sides. This theorem is not only pivotal in theoretical mathematics but is also applicable in numerous real-life situations, such as in construction, navigation, and computer graphics.
In this article, we will explore Pythagorean Theorem word problems, provide some worksheets to practice these concepts, and give answers to enhance understanding.
Understanding the Pythagorean Theorem
What is the Pythagorean Theorem? 🤔
The Pythagorean Theorem applies to right-angled triangles only. According to this theorem, 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 relationship allows us to find an unknown side length if the lengths of the other two sides are known.
The Importance of Word Problems 📝
Word problems help students apply theoretical knowledge in practical contexts. They encourage critical thinking and problem-solving skills. In this case, Pythagorean Theorem word problems can involve various real-life scenarios, from measuring distances to calculating heights and slopes.
Types of Pythagorean Theorem Word Problems
When dealing with word problems related to the Pythagorean Theorem, one might encounter various scenarios, such as:
- Finding the Length of a Side: Given the lengths of two sides, find the length of the hypotenuse or the missing leg.
- Real-World Applications: Apply the theorem to solve problems about heights, distances, and angles in practical situations.
- Complex Shapes: Use the theorem in conjunction with other geometric principles to find lengths in more complex figures.
Example Word Problems
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Problem 1: A ladder leans against a wall. The foot of the ladder is 3 feet away from the wall, and the ladder is 5 feet long. How high up the wall does the ladder reach?
Solution: We can use the Pythagorean Theorem. Let a be the distance from the wall (3 ft), b be the height on the wall we want to find, and c be the length of the ladder (5 ft). Thus, the equation is:
[ a^2 + b^2 = c^2 ]
[ 3^2 + b^2 = 5^2 ]
[ 9 + b^2 = 25 ]
[ b^2 = 25 - 9 ]
[ b^2 = 16 \Rightarrow b = 4 \text{ ft} ] -
Problem 2: A rectangular garden measures 8 meters by 6 meters. What is the length of the diagonal of the garden?
Solution: Here, the garden forms a right triangle where the sides are 8 meters and 6 meters. Using the Pythagorean Theorem: [ a^2 + b^2 = c^2 ]
[ 8^2 + 6^2 = c^2 ]
[ 64 + 36 = c^2 ]
[ 100 = c^2 \Rightarrow c = 10 \text{ m} ]
Practice Worksheet
Here’s a simple worksheet for practice, featuring a few word problems for students to solve using the Pythagorean Theorem.
| Problem Number | Problem Description |
|---|---|
| 1 | A right triangle has legs measuring 5 cm and 12 cm. What is the length of the hypotenuse? |
| 2 | A boat sails 8 km east and then 6 km north. How far is the boat from its starting point? |
| 3 | A telephone pole is 30 ft tall, and the distance from the base of the pole to the top is 36 ft. How far is the base from the point directly under the top? |
| 4 | A rectangle has a width of 9 meters and a height of 12 meters. What is the diagonal length? |
Answers to Practice Worksheet
Below are the answers to the practice problems listed above.
| Problem Number | Solution |
|---|---|
| 1 | (\sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \text{ cm}) |
| 2 | (\sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \text{ km}) |
| 3 | (\sqrt{30^2 + b^2} = 36) → (900 + b^2 = 1296) → (b^2 = 396) → (b \approx 19.9 \text{ ft}) |
| 4 | (\sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15 \text{ m}) |
Important Note
"Always remember to draw a diagram when solving word problems related to the Pythagorean Theorem. It helps visualize the relationships between the sides of the triangle and makes it easier to set up the equation."
By practicing with these word problems, students can better understand how to apply the Pythagorean Theorem in various contexts. Mastery of this concept is essential for progressing in geometry and for applying mathematical reasoning in real-life situations. Encourage exploration beyond worksheets and into practical scenarios, allowing the theorem to come alive in everyday problem-solving!