what's the area
Geometry
Level
2
A trapezoid has parallel sides of lengths 10 and 15; its two other sides have lengths 3 and 4. Find the area.
The answer is 30.
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1 solution
Draw trapezoid with upper base = 10, lower base = 15, left leg = 3, and right leg = 4.
Draw altitude h from each end of upper base. Label distance from lower left vertex to left altitude as x.
Label distance from lower right vertex to right altitude as y.
Using Pythagorean’s theorem, x^2 + h^2 = 9 or solving for x, x= sqrt(9 - h^2).
And y^2 + h^2 = 16 so y = sqrt(16 - h^2).
(1) Area of trapezoid = .5 (10+15) h = 12.5*h. [Area formula]
(2) Area of trapezoid = 10 h + .5 x h + .5 y*h. [Sum of areas of two triangles and rectangle]
Substitute for x and y, set eqn. 1 = eqn. 2 and solve for h.
Multiply both sides by 2/h and subtract 20*h from both sides
Simplifying 5 = sqrt(9-h^2) + sqrt(16-h^2). When I graph this h=2.4
Area = .5 (10+15) 2.4 = 30.