If The Equation 18 2 + B 2 = 30 2 18^2 + B^2 = 30^2 1 8 2 + B 2 = 3 0 2 Is Found To Be True, What Do We Know About The Triangle?A. The Triangle Is A Right Triangle With A Missing Side Of 34.99.B. The Triangle Is A Right Triangle, And The Legs Are 30 And 24.C. The Triangle Is A

The Pythagorean theorem is a fundamental concept in geometry that describes the relationship between the lengths of the sides of a right-angled triangle. The theorem 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. This theorem is often expressed mathematically as $a^2+b^2=c^2$a^2 + b^2 = c^2}, where ${a$ and $b$b} are the lengths of the two sides that form the right angle, and ${c$ is the length of the hypotenuse.

The Given Equation and Its Implications

In the given equation ${18^2 + b^2 = 30^2}$, we are asked to determine the properties of a triangle that satisfies this equation. To begin, let's analyze the equation and understand its implications. The equation can be rewritten as ${324 + b^2 = 900}$. By subtracting 324 from both sides of the equation, we get ${b^2 = 576}$. Taking the square root of both sides, we find that ${b = 24}$.

Properties of the Triangle

Now that we have found the value of ${b}$, we can use this information to determine the properties of the triangle. Since the equation ${18^2 + b^2 = 30^2}$ is true, we know that the triangle is a right triangle. This is because the Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

The Length of the Hypotenuse

To find the length of the hypotenuse, we can use the Pythagorean theorem. Since we know the lengths of the other two sides, we can plug these values into the theorem and solve for the length of the hypotenuse. In this case, the length of the hypotenuse is equal to the square root of the sum of the squares of the lengths of the other two sides. Therefore, the length of the hypotenuse is ${\sqrt{18^2 + 24^2} = \sqrt{324 + 576} = \sqrt{900} = 30}$.

Conclusion

In conclusion, if the equation ${18^2 + b^2 = 30^2}$ is true, we know that the triangle is a right triangle with legs of length 18 and 24, and a hypotenuse of length 30. This is because the Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

The Final Answer

Therefore, the correct answer is:

• The triangle is a right triangle with legs of length 18 and 24, and a hypotenuse of length 30.

The Pythagorean theorem is a fundamental concept in geometry that describes the relationship between the lengths of the sides of a right-angled triangle. In this article, we will answer some frequently asked questions about the Pythagorean theorem and right triangles.

Q: What is the Pythagorean theorem?

A: The Pythagorean theorem is a mathematical formula that describes the relationship between the lengths of the sides of a right-angled triangle. It 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 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. You can then plug these values into the theorem and solve for the length of the hypotenuse. The formula is: ${c^2 = a^2 + b^2}$, where ${c}$ is the length of the hypotenuse, and ${a}$ and ${b}$ are the lengths of the other two sides.

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

A: To use the Pythagorean theorem to find the length of a side, you need to know the lengths of the other two sides. You can then plug these values into the theorem and solve for the length of the side. The formula is: ${a^2 = c^2 - b^2}$, where ${a}$ is the length of the side you want to find, ${c}$ is the length of the hypotenuse, and ${b}$ is the length of the other side.

Q: What is a right triangle?

A: A right triangle is a triangle that has one right angle (90 degrees). The side opposite the right angle is called the hypotenuse, and the other two sides are called the legs.

Q: What is the difference between a right triangle and an oblique triangle?

A: A right triangle is a triangle that has one right angle (90 degrees). An oblique triangle is a triangle that does not have a right angle.

Q: Can I use the Pythagorean theorem to find the length of a side in an oblique triangle?

A: No, you cannot use the Pythagorean theorem to find the length of a side in an oblique triangle. The Pythagorean theorem only works for right triangles.

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

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

• Building design and construction
• Physics and engineering
• Navigation and surveying
• Computer graphics and game development

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-angled triangle. It has many real-world applications and is used in a variety of fields, including building design and construction, physics and engineering, navigation and surveying, and computer graphics and game development.

Q: What is the given equation?

A: The given equation is ${18^2 + b^2 = 30^2}$.

Q: What does the given equation represent?

A: The given equation represents a right triangle with legs of length 18 and an unknown side of length ${b}$.

Q: How do I solve the given equation for ${b}$?

A: To solve the given equation for ${b}$, you can subtract ${18^2}$ from both sides of the equation and then take the square root of both sides.

Q: What is the value of ${b}$?

A: The value of ${b}$ is 24.

Q: What does the value of ${b}$ represent?

A: The value of ${b}$ represents the length of the unknown side of the right triangle.

Conclusion

In conclusion, the given equation ${18^2 + b^2 = 30^2}$ represents a right triangle with legs of length 18 and an unknown side of length ${b}$. The value of ${b}$ is 24, which represents the length of the unknown side of the right triangle.