Which Equation Can Be Used To Find $x$, The Length Of The Hypotenuse Of The Right Triangle?A. $16 + 63 = X$ B. $ 16 2 + 63 = X 16^2 + 63 = X 1 6 2 + 63 = X [/tex] C. $(16 + 63)^2 = X^2$ D. $16^2 + 63^2 = X^2$

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Understanding the Basics of Right Triangles

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 always the longest side of the triangle. In this article, we will explore the different equations 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 (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:

a2+b2=c2a^2 + b^2 = c^2

Applying the Pythagorean Theorem to the Given Options

Now that we have a basic understanding of the Pythagorean Theorem, let's apply it to the given options to determine which equation can be used to find the length of the hypotenuse of the right triangle.

Option A: $16 + 63 = x$

This option is incorrect because it does not take into account the relationship between the lengths of the sides of the right triangle. The Pythagorean Theorem 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, not the sum of the lengths of the other two sides.

Option B: $16^2 + 63 = x$

This option is also incorrect because it does not take into account the relationship between the lengths of the sides of the right triangle. The Pythagorean Theorem 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, not the sum of the squares of one side and the length of the other side.

Option C: $(16 + 63)^2 = x^2$

This option is incorrect because it does not take into account the relationship between the lengths of the sides of the right triangle. The Pythagorean Theorem 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, not the square of the sum of the lengths of the other two sides.

Option D: $16^2 + 63^2 = x^2$

This option is correct because it takes into account the relationship between the lengths of the sides of the right triangle. The Pythagorean Theorem 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, which is exactly what this option represents.

Conclusion

In conclusion, the correct equation to find the length of the hypotenuse of the right triangle is:

162+632=x216^2 + 63^2 = x^2

This equation represents the Pythagorean Theorem, which describes the relationship between the lengths of the sides of a right triangle. By using this equation, we can find the length of the hypotenuse of a right triangle given the lengths of the other two sides.

Real-World Applications

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 perpendicular and at the correct angles.
  • Surveying: Surveyors use the Pythagorean Theorem to measure the distance between two points and to calculate the height of a building or a mountain.
  • 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 bridges, roads, and other infrastructure projects.

Final Thoughts

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). This can be expressed mathematically as:

a2+b2=c2a^2 + b^2 = c^2

Q: What is a right triangle?

A: 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 always the longest side of the triangle.

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). You can then plug these values into the equation:

a2+b2=c2a^2 + b^2 = c^2

and solve for c.

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 use other methods to find the length of the hypotenuse, such as using trigonometry or geometry.

Q: Can I use the Pythagorean Theorem to find the length of the other two sides?

A: Yes, you can use the Pythagorean Theorem to find the length of the other two sides. If you know the length of the hypotenuse (c) and one of the other sides (a or b), you can plug these values into the equation:

a2+b2=c2a^2 + b^2 = c^2

and solve for the unknown side.

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

A: 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 perpendicular and at the correct angles.
  • Surveying: Surveyors use the Pythagorean Theorem to measure the distance between two points and to calculate the height of a building or a mountain.
  • 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 bridges, roads, and other infrastructure projects.

Q: Can I use the Pythagorean Theorem to find the area of a triangle?

A: No, the Pythagorean Theorem is used to find the length of the hypotenuse of a right triangle, not the area of a triangle. To find the area of a triangle, you need to use other formulas, such as the formula for the area of a triangle:

A=12bhA = \frac{1}{2}bh

where A is the area, b is the base, and h is the height.

Q: Can I use the Pythagorean Theorem to find the perimeter of a triangle?

A: No, the Pythagorean Theorem is used to find the length of the hypotenuse of a right triangle, not the perimeter of a triangle. To find the perimeter of a triangle, you need to add up the lengths of all three sides.

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: Make sure to square the lengths of the sides before plugging them into the equation.
  • Not using the correct formula: Make sure to use the correct formula for the Pythagorean Theorem, which is:

a2+b2=c2a^2 + b^2 = c^2

  • Not solving for the correct variable: Make sure to solve for the correct variable, which is usually the length of the hypotenuse (c).

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

A: No, the Pythagorean Theorem is only used to find the length of the hypotenuse of a right triangle. If you have a non-right triangle, you need to use other methods to find the length of the sides, such as using trigonometry or geometry.