Crafts. Rose’s garden is in the shape of a trapezoid?

If the height of the trapezoid


is 16 m, one base is 20 m, and the area is 224 m2, find the length of the other base.





I am not very good at this. Please help.

Crafts. Rose’s garden is in the shape of a trapezoid?
A trapezoid is basically a triangle with a section cut off at a certain height.





Triangles have the property that, if the apex moves around but still stays the same right-angle height above the base, the area of the triangle is still the same, whether the apex is just above the base or 5,000 km to the side of the base but still at the same right-angle height (parallel).





Imagine a triangle cut in two to form a trapezoid and a smaller triangle.





Trapezoid area = Area of large triangle - area of small triangle.





If the base is b and the other base of the trapezoid (above the first and parallel) is a smaller value a, and if H is the large-triangle height (always at right-angles to the base b) and h the trapezoid height, then subtracting the area of the small triangle from the bigger one,





Trapezoid area = 1/2 bH - 1/2 a (H-h) = 1/2 (b-a)H + 1/2 (ah)





but as the height increases, the width of a triangle decreases in proportion, so we know





a= b(H-h)/H (a consequence of the geometry of similar triangles, if the height increases, the base increases proportionally)





Area = 1/2 bH - 1/2 aH + 1/2 ah





= 1/2 bH - 1/2 [b(H-h)/H]*H + 1/2ah





= 1/2 bH - 1/2 [b(H-h)] + 1/2ah





= 1/2 bH - 1/2 bH + 1/2 bh + 1/2 ah





= 1/2 (a+b)h.





(here all H-terms cancel out, meaning we don't need to define a trapezoid in terms of the parent triangle any more but only in its own terms b, a and h.)





Trapezoid area = 1/2 (a+b)h





But we know area is 224 m^2, b is 20 and h is 16.





224 = 1/2 (a+20)*16


224/16 = 1/2 (a+20)


224/16*2 = a+20


224/16*2 - 20 = a


8 = a





So the other base, 16 meters above the 20-meter baseline, is 8 meters wide.





Another way to think about it is that if you rotate the sides of the trapezoid at a mid-point so the sides are now at right angles, you have removed a small triangle from the bottom halves of each side to re-fit it to the top, so geometrically you prove that the area of the trapezoid equals the area of the rectangle of the same height and a width which is the average of the two bases.





(20+8)/2 * h = 14*16 = 224 m^2.

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