+4622 In trapezoid ABCD the lengths of the bases AB and CD are 8 and 17 respectively. The legs of the trapezoid are extended beyond A and B to meet at point . What is the ratio of the area of triangle EAB to the area of trapezoid ABCD? Express your answer as a common fraction.
+118687 Hi Tetre,
The word trapezoid means different things in different countries, could you plase give a definition that we should use?
I'll just tell what I think you mean.
AB and DC are parallel.
This means that AEB and DEC are similar triangles
so
the ratio of the height of ABE and DEC will be 8:17
so Area of AEB = 0.5*8*8k and area of DEC = 0.5*17* 17k
and area of ABCD = 0.5*17* 17k - 0.5*8*8k
ratio of the area of triangle EAB to the area of trapezoid ABCD
\(=\frac{0.5*8*8k}{0.5*17*17k\;-\;0.5*8*8k}\\ =\frac{8*8k}{17*17k\;-\;8*8k}\\ =\frac{8*8}{289\;-\;64}\\ =\frac{64}{225}\\\)
+129899
Since the two triangles created are similar, each dimension of the larger triangle created will be (17/8) that of the smaller triangle
Call h the height of the smaller triangle......so its area = (1/2)*8*h = 4h (1)
And the area of the larger triangle = (1/2)* 17 * (17/8) h = (289 / 16)h (2)
So the area of the trapezoid = (2) - (1) = (289/16) h - 4h = [ 289 - 64 ] / 16 * h =
[225 / 16] h
So.....the ratio of the smaller triangle to the trapezoid = [4h] / ( [ 225/ 16] h ) = 64 / 225
Just as Melody found....!!!!
