Not squaring the lengths of the sides

Oct 13, 2025 by leolavanderia.com.br 15 views

Which Equation Can Be Used to Find the Length of the Hypotenuse of a Right Triangle?

Understanding the Basics of Right Triangles

In mathematics, a right triangle is a triangle with one angle that is 90 degrees. The side opposite the right angle is called the hypotenuse, and it is the longest side of the triangle. When dealing with right triangles, it is often necessary to find the length of the hypotenuse. This can be achieved using the Pythagorean theorem, which is a fundamental concept in geometry.

The Pythagorean Theorem

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). This can be expressed mathematically as:

a^2 + b^2 = c^2

Applying the Pythagorean Theorem to the Given Problem

In the given problem, we are asked to find the length of the hypotenuse (x) of a right triangle, given the lengths of the other two sides (18 and 24). To find the length of the hypotenuse, we can use the Pythagorean theorem.

Option A: 18 + 24 = x

This option is incorrect because it simply adds the lengths of the two sides together, rather than squaring them and adding them together. This is not a valid way to find the length of the hypotenuse.

Option B: 18^2 + 24 = x

This option is also incorrect because it does not square the lengths of the two sides. Instead, it squares only one of the sides and adds the result to the other side. This is not a valid way to find the length of the hypotenuse.

Option C: (18 + 24)^2 = x^2

This option is incorrect because it squares the sum of the lengths of the two sides, rather than squaring each side individually and adding them together. This is not a valid way to find the length of the hypotenuse.

Option D: 18^2 + 24^2 = x^2

This option is correct because it squares the lengths of the two sides and adds them together, which is in line with the Pythagorean theorem. To find the length of the hypotenuse (x), we can take the square root of both sides of the equation:

x^2 = 18^2 + 24^2 x^2 = 324 + 576 x^2 = 900 x = √900 x = 30

Therefore, the length of the hypotenuse (x) is 30.

Conclusion

In conclusion, the correct equation to find the length of the hypotenuse of a right triangle is 18^2 + 24^2 = x^2. This equation is based on the Pythagorean theorem, which states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. By squaring the lengths of the two sides and adding them together, we can find the length of the hypotenuse.

Real-World Applications of the Pythagorean Theorem

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

Building Design: Architects use the Pythagorean theorem to design buildings and ensure that the walls and floors are properly aligned.
Surveying: Surveyors use the Pythagorean theorem to measure the distance between two points and determine the location of a property.
Physics: Physicists use the Pythagorean theorem to calculate the distance and velocity of objects in motion.
Engineering: Engineers use the Pythagorean theorem to design and build structures such as bridges and buildings.

Common Mistakes to Avoid

When using the Pythagorean theorem, there are several common mistakes to avoid:

Not squaring the lengths of the sides: Make sure to square the lengths of the two sides before adding them together.
Not adding the squares together: Make sure to add the squares of the two sides together, rather than squaring the sum of the sides.
Not taking the square root: Make sure to take the square root of both sides of the equation to find the length of the hypotenuse.

Practice Problems

To practice using the Pythagorean theorem, try the following problems:

Find the length of the hypotenuse of a right triangle with sides of length 15 and 20.
Find the length of the hypotenuse of a right triangle with sides of length 10 and 25.
Find the length of the hypotenuse of a right triangle with sides of length 12 and 16.

Conclusion

In conclusion, the Pythagorean theorem is a fundamental concept in geometry that can be used to find the length of the hypotenuse of a right triangle. By squaring the lengths of the two sides and adding them together, we can find the length of the hypotenuse. The Pythagorean theorem has many real-world applications and is an essential tool for architects, surveyors, physicists, and engineers.
Frequently Asked Questions About the Pythagorean Theorem

Q: What is the Pythagorean theorem?

A: The Pythagorean theorem is a fundamental concept in geometry that states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). This can be expressed mathematically as:

a^2 + b^2 = c^2

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

A: To use the Pythagorean theorem to find the length of the hypotenuse, you need to square the lengths of the two sides and add them together. Then, take the square root of the result to find the length of the hypotenuse.

Q: What if I have a right triangle with sides of length 10 and 15? How do I find the length of the hypotenuse?

A: To find the length of the hypotenuse, you need to square the lengths of the two sides and add them together. Then, take the square root of the result to find the length of the hypotenuse.

10^2 + 15^2 = 100 + 225 = 325 √325 ≈ 18.03

Therefore, the length of the hypotenuse is approximately 18.03.

Q: What if I have a right triangle with sides of length 12 and 16? How do I find the length of the hypotenuse?

A: To find the length of the hypotenuse, you need to square the lengths of the two sides and add them together. Then, take the square root of the result to find the length of the hypotenuse.

12^2 + 16^2 = 144 + 256 = 400 √400 = 20

Therefore, the length of the hypotenuse is 20.

Q: Can I use the Pythagorean theorem to find the length of the sides of a right triangle?

A: Yes, you can use the Pythagorean theorem to find the length of the sides of a right triangle. If you know the length of the hypotenuse and one of the other sides, you can use the theorem to find the length of the other side.

Q: What if I have a right triangle with a hypotenuse of length 20 and one side of length 15? How do I find the length of the other side?

A: To find the length of the other side, you need to square the length of the hypotenuse and the length of the known side, and then subtract the result from the square of the hypotenuse.

20^2 - 15^2 = 400 - 225 = 175 √175 ≈ 13.23

Therefore, the length of the other side is approximately 13.23.

Q: Can I use the Pythagorean theorem to find the length of the hypotenuse of a right triangle if I know the lengths of the other two sides?

A: Yes, you can use the Pythagorean theorem to find the length of the hypotenuse of a right triangle if you know the lengths of the other two sides. Simply square the lengths of the two sides and add them together, and then take the square root of the result to find the length of the hypotenuse.

Q: What if I have a right triangle with sides of length 10 and 15? How do I find the length of the hypotenuse?

A: To find the length of the hypotenuse, you need to square the lengths of the two sides and add them together. Then, take the square root of the result to find the length of the hypotenuse.

10^2 + 15^2 = 100 + 225 = 325 √325 ≈ 18.03

Therefore, the length of the hypotenuse is approximately 18.03.

Conclusion

In conclusion, the Pythagorean theorem is a fundamental concept in geometry that can be used to find the length of the hypotenuse of a right triangle. By squaring the lengths of the two sides and adding them together, and then taking the square root of the result, you can find the length of the hypotenuse. The Pythagorean theorem has many real-world applications and is an essential tool for architects, surveyors, physicists, and engineers.