Certainly! Let's work through the problem step-by-step to find the area [tex]\( A \)[/tex] of a trapezoid given the bases [tex]\( b_1 \)[/tex], [tex]\( b_2 \)[/tex], and the height [tex]\( h \)[/tex].
### Step-by-Step Solution
1. Understanding the Area Formula of a Trapezoid:
The area [tex]\( A \)[/tex] of a trapezoid can be calculated using the formula:
[tex]\[
A = \frac{1}{2} \times (b_1 + b_2) \times h
\][/tex]
where:
- [tex]\( b_1 \)[/tex] is the length of the first base.
- [tex]\( b_2 \)[/tex] is the length of the second base.
- [tex]\( h \)[/tex] is the height of the trapezoid (the perpendicular distance between the bases).
2. Simplifying the Formula:
We can rewrite the formula as:
[tex]\[
A = \frac{(b_1 + b_2)}{2} \times h
\][/tex]
3. Expression for the Area:
Plugging [tex]\( b_1 \)[/tex], [tex]\( b_2 \)[/tex], and [tex]\( h \)[/tex] into this formula, we obtain:
[tex]\[
A = h \times \left( \frac{b_1 + b_2}{2} \right)
\][/tex]
4. Final Expression:
Therefore, the area [tex]\( A \)[/tex] of the trapezoid can be expressed as:
[tex]\[
A = h \times \left( \frac{b_1}{2} + \frac{b_2}{2} \right)
\][/tex]
So, the area of the trapezoid in terms of [tex]\( b_1 \)[/tex], [tex]\( b_2 \)[/tex], and [tex]\( h \)[/tex] is:
[tex]\[
A = h \left( \frac{b_1}{2} + \frac{b_2}{2} \right)
\][/tex]
