The trapezoid in Exam Figure 6 has equal nonparallel sides. The upper base is 6, the lower base is 16, and the diagonal is 12. What is the altitude of the trapezoid? Round your answer to one decimal place.
To find the altitude of the trapezoid, you can use the formula:
Altitude = (2 * Area) / (Sum of the bases)
First, let's find the area of the trapezoid.
The formula for the area of a trapezoid is given by:
Area = (1/2) * (Sum of the bases) * Altitude
Let's denote the altitude as 'h'.
We are given the upper base as 6, the lower base as 16, and the diagonal as 12.
We can see that the diagonal, together with the upper and lower bases, creates two right triangles within the trapezoid.
Using the Pythagorean theorem, we can find the height 'h' of the right triangle formed by the diagonal and the altitude.
We have:
(Altitude)^2 + (6 - 16)^2 = (12)^2
Simplifying:
(Altitude)^2 + (-10)^2 = 144
(Altitude)^2 + 100 = 144
(Altitude)^2 = 144 - 100
(Altitude)^2 = 44
Altitude = sqrt(44)
Altitude ≈ 6.63 (rounded to two decimal places)
Now that we have the value of the altitude, we can calculate the area.
Area = (1/2) * (Sum of the bases) * Altitude
Area = (1/2) * (6 + 16) * 6.63
Area = (1/2) * 22 * 6.63
Area ≈ 146.43 (rounded to two decimal places)
Finally, we can find the altitude of the trapezoid using the formula mentioned earlier:
Altitude = (2 * Area) / (Sum of the bases)
Altitude = (2 * 146.43) / (6 + 16)
Altitude = (292.86) / (22)
Altitude ≈ 13.31
Therefore, the altitude of the trapezoid is approximately 13.3 when rounded to one decimal place.