Which Equation Can Be Used To Find \[$ X \$\], The Length Of The Hypotenuse Of The Right Triangle?A. \[$ 18 + 24 = X \$\]B. \[$ 18^2 + 24 = X \$\]C. \[$ (18 + 24)^2 = X^2 \$\]D. \[$ 18^2 + 24^2 = X^2 \$\]

by ADMIN 205 views

Introduction

In geometry, 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. Finding the length of the hypotenuse is an important problem in mathematics, and it has many practical applications in fields such as engineering, physics, and computer science. In this article, we will discuss the equation that can be used to find the length of the hypotenuse of a right triangle.

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 (x) 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:

x2=a2+b2x^2 = a^2 + b^2

Applying the Pythagorean Theorem

To find the length of the hypotenuse of a right triangle, we can use the Pythagorean theorem. Let's consider an example where the lengths of the other two sides are 18 and 24. We can plug these values into the equation to find the length of the hypotenuse.

x2=182+242x^2 = 18^2 + 24^2

Evaluating the Options

Now, let's evaluate the options given in the problem to determine which one is correct.

A. 18 + 24 = x This option is incorrect because it is simply adding the lengths of the other two sides, which is not the correct way to find the length of the hypotenuse.

B. 18^2 + 24 = x This option is also incorrect because it is not squaring the lengths of the other two sides, which is a necessary step in applying the Pythagorean theorem.

C. (18 + 24)^2 = x^2 This option is incorrect because it is squaring the sum of the lengths of the other two sides, which is not the correct way to apply the Pythagorean theorem.

D. 18^2 + 24^2 = x^2 This option is correct because it is squaring the lengths of the other two sides and adding them together, which is the correct way to apply the Pythagorean theorem.

Conclusion

In conclusion, the equation that can be used to find the length of the hypotenuse of a right triangle is:

x2=a2+b2x^2 = a^2 + b^2

This equation is known as the Pythagorean theorem, and it is a fundamental concept in geometry. By applying this theorem, we can find the length of the hypotenuse of a right triangle, which has many practical applications in fields such as engineering, physics, and computer science.

Example Problems

Here are some example problems that illustrate how to use the Pythagorean theorem to find the length of the hypotenuse of a right triangle.

Example 1

Find the length of the hypotenuse of a right triangle with legs of length 15 and 20.

x2=152+202x^2 = 15^2 + 20^2

x2=225+400x^2 = 225 + 400

x2=625x^2 = 625

x=625x = \sqrt{625}

x=25x = 25

Example 2

Find the length of the hypotenuse of a right triangle with legs of length 7 and 24.

x2=72+242x^2 = 7^2 + 24^2

x2=49+576x^2 = 49 + 576

x2=625x^2 = 625

x=625x = \sqrt{625}

x=25x = 25

Example 3

Find the length of the hypotenuse of a right triangle with legs of length 9 and 12.

x2=92+122x^2 = 9^2 + 12^2

x2=81+144x^2 = 81 + 144

x2=225x^2 = 225

x=225x = \sqrt{225}

x=15x = 15

Tips and Tricks

Here are some tips and tricks for using the Pythagorean theorem to find the length of the hypotenuse of a right triangle.

  • Make sure to square the lengths of the other two sides before adding them together.
  • Use the correct formula: $x^2 = a^2 + b^2$
  • Check your work by plugging the values back into the equation to make sure you get the correct answer.
  • Practice, practice, practice! The more you practice using the Pythagorean theorem, the more comfortable you will become with it.

Conclusion

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 (x) 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:

x2=a2+b2x^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 follow these steps:

  1. Identify the lengths of the other two sides of the right triangle.
  2. Square the lengths of the other two sides.
  3. Add the squared lengths together.
  4. Take the square root of the result to find the length of the hypotenuse.

Q: What if I have a right triangle with a hypotenuse of length 10 and one leg of length 6? How do I find the length of the other leg?

A: To find the length of the other leg, you can use the Pythagorean theorem. Let's call the length of the other leg "b". We know that the length of the hypotenuse (x) is 10, and the length of one leg (a) is 6. We can plug these values into the equation:

102=62+b210^2 = 6^2 + b^2

100=36+b2100 = 36 + b^2

b2=64b^2 = 64

b=64b = \sqrt{64}

b=8b = 8

Q: Can I use the Pythagorean theorem to find the length of a side of a triangle that is not a right triangle?

A: No, the Pythagorean theorem only applies to right triangles. If you have a triangle that is not a right triangle, you will need to use a different method to find the length of a side.

Q: How do I know if a triangle is a right triangle?

A: To determine if a triangle is a right triangle, you can use the following methods:

  1. Check if one of the angles is 90 degrees.
  2. Use the Pythagorean theorem to check if the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
  3. Draw a diagram of the triangle and check if it is a right triangle.

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

A: No, the Pythagorean theorem only applies to triangles. If you have a shape that is not a triangle, you will need to use a different method to find the length of a side.

Q: How do I use the Pythagorean theorem to find the length of a side of a triangle that is a right triangle, but the lengths of the sides are not given?

A: To use the Pythagorean theorem to find the length of a side of a right triangle when the lengths of the sides are not given, you will need to use the following steps:

  1. Draw a diagram of the triangle and label the sides.
  2. Use the Pythagorean theorem to set up an equation.
  3. Solve the equation to find the length of the side.

Q: Can I use the Pythagorean theorem to find the length of a side of a triangle that is a right triangle, but the lengths of the sides are given in terms of variables?

A: Yes, you can use the Pythagorean theorem to find the length of a side of a right triangle when the lengths of the sides are given in terms of variables. You will need to substitute the variables into the equation and solve for the length of the side.

Conclusion

In 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 using the Pythagorean theorem, you can find the length of the hypotenuse of a right triangle, and you can also use it to find the length of a side of a right triangle when the lengths of the sides are given. Remember to square the lengths of the other two sides before adding them together, and use the correct formula: $x^2 = a^2 + b^2$. With practice and patience, you will become proficient in using the Pythagorean theorem to find the length of a side of a right triangle.