Question
the longer diagonal of a right trapezoid is 13 cm. the longer base is 12 cm. find the area of the trapezoid the shorter base is 8cm
Answers
GPT-4o mini
To find the area of the trapezoid, we can use the formula for the area of a trapezoid, which is:
\[
\text{Area} = \frac{1}{2} \times (b_1 + b_2) \times h
\]
where \( b_1 \) and \( b_2 \) are the lengths of the two bases, and \( h \) is the height of the trapezoid.
In this case, we have:
- The longer base \( b_1 = 12 \) cm
- The shorter base \( b_2 = 8 \) cm
To find the height \( h \), we can use the Pythagorean theorem. In a right trapezoid, the height and the bases form a right triangle with the diagonal. The length of the diagonal is given as 13 cm.
Let's denote:
- The height of the trapezoid as \( h \).
- The horizontal distance (the difference between the two bases) as the base of the right triangle.
The difference between the longer base and the shorter base is:
\[
b_1 - b_2 = 12 - 8 = 4 \text{ cm}
\]
Now we have a right triangle with one leg being the height \( h \), the other leg being \( 4 \) cm, and the hypotenuse being \( 13 \) cm. We can use the Pythagorean theorem:
\[
h^2 + 4^2 = 13^2
\]
Calculating \( 13^2 \) and \( 4^2 \):
\[
h^2 + 16 = 169
\]
Now, solving for \( h^2 \):
\[
h^2 = 169 - 16 = 153
\]
Taking the square root to find \( h \):
\[
h = \sqrt{153} \approx 12.37 \text{ cm}
\]
Now that we have the height \( h \), we can substitute \( b_1 \), \( b_2 \), and \( h \) into the area formula:
\[
\text{Area} = \frac{1}{2} \times (12 + 8) \times \sqrt{153}
\]
Calculating:
\[
\text{Area} = \frac{1}{2} \times 20 \times \sqrt{153}
\]
\[
\text{Area} = 10 \times \sqrt{153} \approx 10 \times 12.37 \approx 123.7 \text{ cm}^2
\]
Thus, the area of the trapezoid is approximately:
\[
\text{Area} \approx 123.7 \, \text{cm}^2
\]
\[
\text{Area} = \frac{1}{2} \times (b_1 + b_2) \times h
\]
where \( b_1 \) and \( b_2 \) are the lengths of the two bases, and \( h \) is the height of the trapezoid.
In this case, we have:
- The longer base \( b_1 = 12 \) cm
- The shorter base \( b_2 = 8 \) cm
To find the height \( h \), we can use the Pythagorean theorem. In a right trapezoid, the height and the bases form a right triangle with the diagonal. The length of the diagonal is given as 13 cm.
Let's denote:
- The height of the trapezoid as \( h \).
- The horizontal distance (the difference between the two bases) as the base of the right triangle.
The difference between the longer base and the shorter base is:
\[
b_1 - b_2 = 12 - 8 = 4 \text{ cm}
\]
Now we have a right triangle with one leg being the height \( h \), the other leg being \( 4 \) cm, and the hypotenuse being \( 13 \) cm. We can use the Pythagorean theorem:
\[
h^2 + 4^2 = 13^2
\]
Calculating \( 13^2 \) and \( 4^2 \):
\[
h^2 + 16 = 169
\]
Now, solving for \( h^2 \):
\[
h^2 = 169 - 16 = 153
\]
Taking the square root to find \( h \):
\[
h = \sqrt{153} \approx 12.37 \text{ cm}
\]
Now that we have the height \( h \), we can substitute \( b_1 \), \( b_2 \), and \( h \) into the area formula:
\[
\text{Area} = \frac{1}{2} \times (12 + 8) \times \sqrt{153}
\]
Calculating:
\[
\text{Area} = \frac{1}{2} \times 20 \times \sqrt{153}
\]
\[
\text{Area} = 10 \times \sqrt{153} \approx 10 \times 12.37 \approx 123.7 \text{ cm}^2
\]
Thus, the area of the trapezoid is approximately:
\[
\text{Area} \approx 123.7 \, \text{cm}^2
\]