An isosceles trapezoid has base angle 45 degrees and bases of length 12 and 32. Find its area.
To find the area of an isosceles trapezoid, you can use the formula:
Area = (1/2) * (base1 + base2) * height
In this case, the bases are 12 and 32, and we need to find the height of the trapezoid.
To find the height, we can use the Pythagorean theorem since we have a right triangle formed by the base, the height, and the angle between them.
Let's label the height as "h" and the leg of the right triangle as "x". Since the angle between the base and the height is 45 degrees, we can set up the following equation:
tan(45) = h / x
The tangent of 45 degrees is 1, so the equation simplifies to:
1 = h / x
Rearranging the equation, we get:
h = x
Now, we can use the Pythagorean theorem:
x^2 + h^2 = base1^2
Since we know that the base1 is 12, we can substitute:
x^2 + h^2 = 12^2
x^2 + h^2 = 144
But we also know that h = x, so we can substitute:
x^2 + x^2 = 144
2x^2 = 144
x^2 = 72
x = sqrt(72)
Now that we know x, we can use it to find the height (h) and the area of the trapezoid.
h = x = sqrt(72)
And finally, we can use the area formula:
Area = (1/2) * (base1 + base2) * height
Area = (1/2) * (12 + 32) * sqrt(72)
Area = (1/2) * 44 * sqrt(72)
Area = 22 * sqrt(72)
Therefore, the area of the isosceles trapezoid is 22 * sqrt(72).