Applying the Pythagorean Theorem to the Given Triangle

Oct 13, 2025 by recoveryshake.com 43 views

Introduction

In geometry, a right triangle is a triangle in which one of the angles is a right angle, i.e., 90 degrees. To determine if a triangle is a right triangle, we can use the Pythagorean theorem, which states that in a right-angled 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.

The Pythagorean Theorem

The Pythagorean theorem is a fundamental concept in geometry and trigonometry. It is named after the ancient Greek philosopher and mathematician Pythagoras, who is credited with its discovery. The theorem is stated as follows:

a^2 + b^2 = c^2

where a and b are the lengths of the two sides that form the right angle, and c is the length of the hypotenuse.

Applying the Pythagorean Theorem to the Given Triangle

In this problem, we are given a triangle with sides of lengths 32, 60, and 68. To determine if this triangle is a right triangle, we can use the Pythagorean theorem. We will assume that the side of length 68 is the hypotenuse, since it is the longest side.

Step 1: Square the Lengths of the Two Sides

First, we need to square the lengths of the two sides that form the right angle. We have:

a^2 = 32^2 = 1024
b^2 = 60^2 = 3600

Step 2: Calculate the Square of the Hypotenuse

Next, we need to calculate the square of the hypotenuse. We have:

c^2 = 68^2 = 4624

Step 3: Check if the Pythagorean Theorem is Satisfied

Now, we need to check if the Pythagorean theorem is satisfied. We have:

a^2 + b^2 = 1024 + 3600 = 4624
c^2 = 4624

Since a^2 + b^2 = c^2, we can conclude that the Pythagorean theorem is satisfied.

Conclusion

Based on the Pythagorean theorem, we can conclude that the triangle with sides of lengths 32, 60, and 68 is a right triangle. The side of length 68 is the hypotenuse, and the sides of lengths 32 and 60 form the right angle.

Answer

The answer is: A. True

Discussion

This problem is a classic example of how to use the Pythagorean theorem to determine if a triangle is a right triangle. The theorem is a powerful tool in geometry and trigonometry, and it has many applications in real-world problems. In this case, we used the theorem to determine if a triangle with specific side lengths is a right triangle.

Real-World Applications

The Pythagorean theorem has many real-world applications, including:

Building design: Architects use the theorem to determine the height of buildings and the length of shadows.
Surveying: Surveyors use the theorem to determine the distance between two points on the Earth's surface.
Physics: Physicists use the theorem to calculate the distance and velocity of objects in motion.

Conclusion

Q: What is a right triangle?

A: A right triangle is a triangle in which one of the angles is a right angle, i.e., 90 degrees.

Q: What is the Pythagorean theorem?

A: The Pythagorean theorem is a fundamental concept in geometry and trigonometry that states that in a right-angled 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.

Q: How do I use the Pythagorean theorem to determine if a triangle is a right triangle?

A: To use the Pythagorean theorem, you need to square the lengths of the two sides that form the right angle, add them together, and check if the result is equal to the square of the length of the hypotenuse.

Q: What are the steps to follow when using the Pythagorean theorem?

A: The steps to follow are:

Square the lengths of the two sides that form the right angle.
Add the squared lengths together.
Check if the result is equal to the square of the length of the hypotenuse.

Q: What is the formula for the Pythagorean theorem?

A: The formula for the Pythagorean theorem is:

a^2 + b^2 = c^2

where a and b are the lengths of the two sides that form the right angle, and c is the length of the hypotenuse.

Q: Can I use the Pythagorean theorem to find the length of the hypotenuse?

A: Yes, you can use the Pythagorean theorem to find the length of the hypotenuse. Simply rearrange the formula to solve for c:

c = sqrt(a^2 + b^2)

Q: Can I use the Pythagorean theorem to find the length of one of the other sides?

A: Yes, you can use the Pythagorean theorem to find the length of one of the other sides. Simply rearrange the formula to solve for a or b:

a = sqrt(c^2 - b^2) b = sqrt(c^2 - a^2)

Q: What are some real-world applications of the Pythagorean theorem?

A: Some real-world applications of the Pythagorean theorem include:

Building design: Architects use the theorem to determine the height of buildings and the length of shadows.
Surveying: Surveyors use the theorem to determine the distance between two points on the Earth's surface.
Physics: Physicists use the theorem to calculate the distance and velocity of objects in motion.

Q: Can I use the Pythagorean theorem to solve problems involving right triangles with more than two sides?

A: Yes, you can use the Pythagorean theorem to solve problems involving right triangles with more than two sides. However, you will need to use the theorem multiple times to find the length of each side.

Q: What are some common mistakes to avoid when using the Pythagorean theorem?

A: Some common mistakes to avoid when using the Pythagorean theorem include:

Not squaring the lengths of the sides correctly
Not adding the squared lengths together correctly
Not checking if the result is equal to the square of the length of the hypotenuse

Conclusion

In conclusion, the Pythagorean theorem is a fundamental concept in geometry and trigonometry that can be used to determine if a triangle is a right triangle. By following the steps outlined in this article, you can use the theorem to solve problems involving right triangles and find the length of the hypotenuse or one of the other sides.