Question 1102095: The longer base of an isosceles trapezoid measures 18 ft. The nonparallel sides measure 11 ft, and the base angles measure 80°.
I was able to find the length of a diagonal which is 19 ft. I need to know how to find the area of the trapezoid.
Thanks friends
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
First let's draw a picture of what this could look like
The longer base is horizontally placed along the bottom, which is the distance from point A to point B. So AB = 18 feet.
The nonparallel sides, AD and BC are both 11 feet long. So AD = 11 and BC = 11.
We don't know the length of CD, which is the length of the other parallel side.
We'll need this to find the area of the trapezoid. For now, let's call it x. So CD = x.
To help find CD, let's drop a perpendicular line straight down from points C and D like shown below
Those perpendicular lines, the dashed vertical ones, drop to points E and F as shown in the figure.
The distance from D to E is some unknown height h.
The distance from A to E is unknown. We'll call it y. So AE = y.
Because we have an isosceles trapezoid, we have symmetry along the vertical center line.
This means FB = y as well. The length of EF is equal to the length of CD. This means EF = x.
Overall, AE+EF+FB = AB translating to y+x+y = 18 or x+2y = 18. Solve for x to get x = -2y+18.
If we knew the value of y, then we can find x.
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To find the value of y, we'll use trigonometry. We'll use the cosine rule
cos(angle) = adjacent/hypotenuse
cos(80) = AE/AD
cos(80) = y/11
11*cos(80) = y
y = 11*cos(80) this is the exact length of AE and FB
y = 1.910130 this is the approximate length of AE and FB (rounded to 6 decimal places)
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We now know the approximate value of y so that makes x to be roughly...
x = -2*y+18
x = -2*1.910130+18
x = 14.17974
So the other parallel side, CD, is roughly 14.17974 feet long
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We're almost at the finish line. We need to find the height h.
We'll use trig again, but this time the sine rule
sin(angle) = opposite/hypotenuse
sin(80) = h/11
h = 11*sin(80)
h = 10.832885
The height of this trapezoid is roughly 10.832885 feet.
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In summary so far, we found
x = 14.17974
h = 10.832885
Which allows us to finally find the area of this trapezoid
Use this area formula
Area of Trapezoid = h*(b1+b2)/2
where,
h = height
b1 & b2 = the parallel bases
So,
Area of Trapezoid = h*(b1+b2)/2
Area of Trapezoid = 10.832885*(14.17974+18)/2
Area of Trapezoid = 174.29971137495
Which is approximate. Due to the approximate nature of the answer, there is likely to be rounding error.
Use more decimal digits in the values of x and h to reduce the rounding error.
Side Note: the units for the answer are "square feet".
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