4)
The median and the height of an isosceles trapezoid are equal in length. One diagonal of the trapezoid has length 12. What is the area of the trapezoid?
See the following image ( not to scale )
Let AD be the diagonal......call the median GJ and the height AE , M
Let GH , IJ = X
And let the top base = B
Then GJ - HI = GH + IJ = 2IJ
So....M - B = GH + IJ = 2X = 2IJ
And since triangles BFD and BIJ are similar...and 2BI = BF and 2BJ = BD....then 2IJ = FD
But 2IJ = 2X
So.....
EF + FD =
B + (2IJ ) =
B + (M - B) =
M =
ED
And because AED is a right angle.. and AED a right triangle......we have that
AE^2 + ED^2 = AD^2
M^2 + M^2 =12^2
2M^2 = 144
M^2 = 72
Note that the area of the trapezoid is (1/2) (height ( sum of the bases) =
(height) (sum of the bases) / 2 =
And ...(sum of the bases) / 2 = the median M ....so....
M * M = M^2 = 72 = area of the trapezoid