Rearrange The Formula For The Area Of A Trapezoid, \[$ A = \frac{1}{2} H(a+b) \$\], To Solve For The Height (\[$ H \$\]).A. \[$ H = \frac{2A}{a+b} \$\]B. \[$ H = \frac{2(e+b)}{A} \$\]C. \[$ H = \frac{A}{2(e+b)}

Feb 27, 2025 by ADMIN 211 views

**Rearranging the Formula for the Area of a Trapezoid: Solving for Height**

Introduction

The area of a trapezoid is a fundamental concept in geometry, and it is often used in various mathematical and real-world applications. The formula for the area of a trapezoid is given by ${$ A = \frac{1}{2} h(a+b) $}$, where ${$ A $}$ is the area, ${$ h $}$ is the height, and ${$ a $}$ and ${$ b $}$ are the lengths of the two parallel sides. In this article, we will focus on rearranging this formula to solve for the height ${$ h $}$.

Understanding the Formula

Before we proceed with rearranging the formula, let's take a closer look at the original formula: ${$ A = \frac{1}{2} h(a+b) $}$. This formula states that the area of a trapezoid is equal to half the product of the height and the sum of the lengths of the two parallel sides. To solve for the height, we need to isolate ${$ h $}$ on one side of the equation.

Rearranging the Formula

To solve for the height, we can start by multiplying both sides of the equation by 2 to eliminate the fraction: ${$ 2A = h(a+b) $}$. Next, we can divide both sides of the equation by ${$ (a+b) $}$ to isolate ${$ h $}$: ${$ h = \frac{2A}{a+b} $}$.

Analyzing the Options

Now that we have rearranged the formula to solve for the height, let's analyze the options provided:

A. ${$ h = \frac{2A}{a+b} $}$

B. ${$ h = \frac{2(e+b)}{A} $}$

C. ${$ h = \frac{A}{2(e+b)} $}$

Option A: Correct Solution

The correct solution is option A: ${$ h = \frac{2A}{a+b} $}$. This solution is derived from the original formula by multiplying both sides by 2 and then dividing both sides by ${$ (a+b) $}$.

Option B: Incorrect Solution

Option B is incorrect because it contains a variable ${$ e $}$ that is not present in the original formula. Additionally, the formula is not correctly rearranged to solve for the height.

Option C: Incorrect Solution

Option C is also incorrect because it contains a variable ${$ e $}$ that is not present in the original formula. Furthermore, the formula is not correctly rearranged to solve for the height.

Conclusion

In conclusion, the correct solution to rearrange the formula for the area of a trapezoid to solve for the height is ${$ h = \frac{2A}{a+b} $}$. This solution is derived from the original formula by multiplying both sides by 2 and then dividing both sides by ${$ (a+b) $}$. The other options provided are incorrect due to the presence of extraneous variables or incorrect rearrangement of the formula.

Additional Tips and Examples

To verify the correctness of the solution, you can substitute the values of ${$ A $}$, ${$ a $}$, and ${$ b $}$ into the formula and check if the result is consistent with the original formula.
The formula for the area of a trapezoid can be used to solve problems involving the calculation of the height of a trapezoid given its area and the lengths of its parallel sides.
The concept of rearranging formulas to solve for different variables is a fundamental skill in mathematics and is used extensively in various mathematical and real-world applications.

Common Mistakes to Avoid

When rearranging formulas, it is essential to ensure that the correct operations are performed to isolate the desired variable.
Be cautious of extraneous variables that may be introduced during the rearrangement process.
Verify the correctness of the solution by substituting values into the formula and checking for consistency with the original formula.

Real-World Applications

The formula for the area of a trapezoid is used in various real-world applications, such as:
- Architecture: to calculate the area of trapezoidal roofs or walls.
- Engineering: to calculate the area of trapezoidal beams or channels.
- Physics: to calculate the area of trapezoidal surfaces in problems involving force and pressure.

Conclusion

Q: What is the formula for the area of a trapezoid?

A: The formula for the area of a trapezoid is ${$ A = \frac{1}{2} h(a+b) $}$, where ${$ A $}$ is the area, ${$ h $}$ is the height, and ${$ a $}$ and ${$ b $}$ are the lengths of the two parallel sides.

Q: How do I rearrange the formula to solve for the height?

A: To solve for the height, you can start by multiplying both sides of the equation by 2 to eliminate the fraction: ${$ 2A = h(a+b) $}$. Next, you can divide both sides of the equation by ${$ (a+b) $}$ to isolate ${$ h $}$: ${$ h = \frac{2A}{a+b} $}$.

Q: What is the correct solution to rearrange the formula for the area of a trapezoid to solve for the height?

A: The correct solution is ${$ h = \frac{2A}{a+b} $}$. This solution is derived from the original formula by multiplying both sides by 2 and then dividing both sides by ${$ (a+b) $}$.

Q: What are some common mistakes to avoid when rearranging formulas?

A: Some common mistakes to avoid when rearranging formulas include:

Introducing extraneous variables that are not present in the original formula.
Not performing the correct operations to isolate the desired variable.
Not verifying the correctness of the solution by substituting values into the formula.

Q: What are some real-world applications of the formula for the area of a trapezoid?

A: The formula for the area of a trapezoid is used in various real-world applications, such as:

Architecture: to calculate the area of trapezoidal roofs or walls.
Engineering: to calculate the area of trapezoidal beams or channels.
Physics: to calculate the area of trapezoidal surfaces in problems involving force and pressure.

Q: How can I verify the correctness of the solution to rearrange the formula for the area of a trapezoid to solve for the height?

A: You can verify the correctness of the solution by substituting values into the formula and checking for consistency with the original formula. For example, if you know the area of a trapezoid is 10 square units, the height is 2 units, and the lengths of the two parallel sides are 3 units and 5 units, you can substitute these values into the formula and check if the result is consistent with the original formula.

Q: What are some additional tips for rearranging formulas?

A: Some additional tips for rearranging formulas include:

Always start by identifying the variable you want to isolate.
Perform the correct operations to isolate the variable.
Verify the correctness of the solution by substituting values into the formula.
Be cautious of extraneous variables that may be introduced during the rearrangement process.

Q: Can I use the formula for the area of a trapezoid to solve problems involving the calculation of the height of a trapezoid given its area and the lengths of its parallel sides?

A: Yes, you can use the formula for the area of a trapezoid to solve problems involving the calculation of the height of a trapezoid given its area and the lengths of its parallel sides. Simply rearrange the formula to solve for the height, and then substitute the given values into the formula to find the height.

Conclusion

In conclusion, rearranging the formula for the area of a trapezoid to solve for the height is a straightforward process that involves multiplying both sides of the equation by 2 and then dividing both sides by ${$ (a+b) $}$. The correct solution is ${$ h = \frac{2A}{a+b} $}$, and the other options provided are incorrect due to the presence of extraneous variables or incorrect rearrangement of the formula.