How do you find the area of a trapezoid when given only the lengths of the sides?

To find the area of a trapezoid when given only the lengths of its sides, we can use the following formula, known as the **Brahmagupta's formula** for a cyclic quadrilateral. However, for a general trapezoid, we need to know the height. If the height is not directly given, we can derive it using the Pythagorean theorem. Here’s the step-by-step process:

### Given:
- Base 1 (\(a\))
- Base 2 (\(b\))
- Side 1 (\(c\))
- Side 2 (\(d\))

### Steps:
1. **Calculate the semi-perimeter (s) of the trapezoid:**
   \[
   s = \frac{a + b + c + d}{2}
   \]

2. **Calculate the height (h) using the formula:**
   \[
   h = \frac{2}{|a - b|} \sqrt{(s-a)(s-b)(s-a-c)(s-a-d)}
   \]

3. **Finally, calculate the area (A) of the trapezoid using the standard trapezoid area formula:**
   \[
   A = \frac{1}{2} \times (a + b) \times h
   \]

### Example Calculation:
Suppose we have a trapezoid with the following side lengths:
- \(a = 8\) units
- \(b = 5\) units
- \(c = 7\) units
- \(d = 9\) units

1. **Calculate the semi-perimeter (s):**
   \[
   s = \frac{8 + 5 + 7 + 9}{2} = \frac{29}{2} = 14.5 \text{ units}
   \]

2. **Calculate the height (h):**
   \[
   h = \frac{2}{|8 - 5|} \sqrt{(14.5 - 8)(14.5 - 5)(14.5 - 7)(14.5 - 9)}
   \]
   \[
   h = \frac{2}{3} \sqrt{(6.5)(9.5)(7.5)(5.5)}
   \]
   \[
   h = \frac{2}{3} \sqrt{6.5 \times 9.5 \times 7.5 \times 5.5}
   \]
   \[
   h = \frac{2}{3} \sqrt{2144.0625} \approx \frac{2}{3} \times 46.29 \approx 30.86 \text{ units}
   \]

3. **Calculate the area (A):**
   \[
   A = \frac{1}{2} \times (8 + 5) \times 30.86 = \frac{1}{2} \times 13 \times 30.86 = 200.59 \text{ square units}
   \]

So, the area of the trapezoid is approximately 200.59 square units.

### Related Questions:
1. What are the steps to derive the height of a trapezoid from its side lengths?
2. How do you use the Pythagorean theorem in the context of trapezoids?
3. What is Brahmagupta's formula and how is it used in geometry?
4. How do you handle trapezoids with non-parallel sides of different lengths?
5. Can you apply the same method to calculate the area of a parallelogram?
6. How do you solve for the area of an isosceles trapezoid?
7. What are some practical applications of calculating the area of a trapezoid?
8. How do you verify if a given quadrilateral is a trapezoid?

**Tip:** When dealing with trapezoids or any other polygons, drawing a clear diagram and labeling all given measurements can significantly help in visualizing and solving the problem correctly.

Learn how to find the area of a trapezoid when given only the lengths of its sides. This comprehensive guide includes the trapezoid area formula and step-by-step instructions using the Pythagorean theorem to derive the height. Suitable for students in Grades 7-9.

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