Why the Correct Option Works

Oct 13, 2025 by miodeya.com 31 views

Understanding the Problem

When dealing with right triangles, it's essential to know how to find the length of the hypotenuse, which is the side opposite the right angle. In this article, we'll explore the different equations that can be used to find the length of the hypotenuse and determine which one is correct.

The Pythagorean Theorem

The Pythagorean Theorem is a fundamental concept in geometry that describes the relationship between the lengths of the sides of a right triangle. The theorem states that 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). Mathematically, this can be expressed as:

c^2 = a^2 + b^2

Analyzing the Options

Now that we understand the Pythagorean Theorem, let's analyze the options provided:

A. $16 + 63 = x$

This option is incorrect because it simply adds the lengths of the two sides, which is not what the Pythagorean Theorem states.

B. $16^2 + 63 = x$

This option is also incorrect because it adds the square of one side to the other side, rather than squaring both sides and adding them together.

C. $(16 + 63)^2 = x^2$

This option is incorrect because it squares the sum of the two sides, rather than squaring each side individually and adding them together.

D. $16^2 + 63^2 = x^2$

This option is correct because it squares both sides and adds them together, which is in line with the Pythagorean Theorem.

Why the Correct Option Works

The correct option, D, works because it follows the Pythagorean Theorem. By squaring both sides and adding them together, we get the square of the length of the hypotenuse, which is what we're trying to find. This equation can be used to find the length of the hypotenuse of a right triangle, as long as we know the lengths of the other two sides.

Real-World Applications

The Pythagorean Theorem has numerous real-world applications, including:

Building design: Architects use the Pythagorean Theorem to calculate the height of buildings and the length of shadows.
Surveying: Surveyors use the Pythagorean Theorem to calculate distances and angles between landmarks.
Physics: Physicists use the Pythagorean Theorem to calculate distances and velocities in problems involving right triangles.

Conclusion

In conclusion, the correct equation to find the length of the hypotenuse of a right triangle is D. $16^2 + 63^2 = x^2$ . This equation follows the Pythagorean Theorem and can be used to find the length of the hypotenuse, as long as we know the lengths of the other two sides. By understanding the Pythagorean Theorem and how to apply it, we can solve a wide range of problems involving right triangles.

Additional Examples

Here are a few additional examples of how to use the Pythagorean Theorem to find the length of the hypotenuse:

Example 1: Find the length of the hypotenuse of a right triangle with legs of length 3 and 4.
Solution: Using the Pythagorean Theorem, we get: c^2 = 3^2 + 4^2 = 9 + 16 = 25. Taking the square root of both sides, we get: c = √25 = 5.
Example 2: Find the length of the hypotenuse of a right triangle with legs of length 5 and 12.
Solution: Using the Pythagorean Theorem, we get: c^2 = 5^2 + 12^2 = 25 + 144 = 169. Taking the square root of both sides, we get: c = √169 = 13.

Common Mistakes

When using the Pythagorean Theorem, there are a few common mistakes to watch out for:

Not squaring both sides: Make sure to square both sides of the equation, rather than just one side.
Not adding the squares: Make sure to add the squares of the two sides, rather than just adding the sides themselves.
Not taking the square root: Make sure to take the square root of both sides, rather than just leaving the equation as is.

Q: What is the Pythagorean Theorem?

A: The Pythagorean Theorem is a fundamental concept in geometry that describes the relationship between the lengths of the sides of a right triangle. It states that 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). Mathematically, this can be expressed as:

c^2 = a^2 + b^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 know the lengths of the other two sides (a and b). Then, you can plug these values into the equation:

c^2 = a^2 + b^2

Solve for c by taking the square root of both sides:

c = √(a^2 + b^2)

Q: What if I don't know the lengths of the other two sides?

A: If you don't know the lengths of the other two sides, you can't use the Pythagorean Theorem to find the length of the hypotenuse. However, you can use other methods, such as trigonometry or geometry, to find the length of the hypotenuse.

Q: Can I use the Pythagorean Theorem to find the length of a side that is not the hypotenuse?

A: Yes, you can use the Pythagorean Theorem to find the length of a side that is not the hypotenuse. However, you need to know the lengths of the other two sides and the length of the side you want to find. Then, you can plug these values into the equation:

a^2 = c^2 - b^2

b^2 = c^2 - a^2

Solve for a or b by taking the square root of both sides:

a = √(c^2 - b^2)

b = √(c^2 - a^2)

Q: What are some common mistakes to watch out for when using the Pythagorean Theorem?

A: Some common mistakes to watch out for when using the Pythagorean Theorem include:

Not squaring both sides
Not adding the squares
Not taking the square root
Using the wrong values for a, b, and c

Q: Can I use the Pythagorean Theorem to find the length of a side in a non-right triangle?

A: No, the Pythagorean Theorem only works for right triangles. If you have a non-right triangle, you can't use the Pythagorean Theorem to find the length of a side.

Q: How do I apply the Pythagorean Theorem in real-world situations?

A: The Pythagorean Theorem has numerous real-world applications, including:

Building design
Surveying
Physics
Engineering

To apply the Pythagorean Theorem in real-world situations, you need to identify the right triangle and the lengths of the sides. Then, you can use the Pythagorean Theorem to find the length of the hypotenuse or other sides.

Q: Can I use the Pythagorean Theorem to find the length of a side in a 3D triangle?

A: No, the Pythagorean Theorem only works for 2D triangles. If you have a 3D triangle, you need to use other methods, such as vector calculus or geometry, to find the length of a side.

Q: How do I prove the Pythagorean Theorem?

A: There are several ways to prove the Pythagorean Theorem, including:

Geometric proof
Algebraic proof
Trigonometric proof

To prove the Pythagorean Theorem, you need to show that the equation c^2 = a^2 + b^2 is true for all right triangles.

Q: Can I use the Pythagorean Theorem to find the length of a side in a triangle with a negative length?

A: No, the Pythagorean Theorem only works for triangles with positive lengths. If you have a triangle with a negative length, you can't use the Pythagorean Theorem to find the length of a side.

Q: How do I use the Pythagorean Theorem to find the length of a side in a triangle with a complex length?

A: The Pythagorean Theorem only works for triangles with real lengths. If you have a triangle with a complex length, you need to use other methods, such as complex analysis or geometry, to find the length of a side.

Conclusion

The Pythagorean Theorem is a fundamental concept in geometry that describes the relationship between the lengths of the sides of a right triangle. By understanding the Pythagorean Theorem and how to apply it, you can solve a wide range of problems involving right triangles. Remember to watch out for common mistakes and to use the Pythagorean Theorem only for right triangles with real lengths.