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Work Shown:

We use the intersecting chords theorem

(AH)*(HG) = (FH)*(HB)

(x-2)*(x+7) = (2x-1)*(4)

x*(x+7) - 2(x+7) = 4(2x-1)

x^2+7x - 2x - 14 = 8x - 4

x^2 + 7x - 2x - 14 - 8x + 4 = 0

x^2 - 3x - 10 = 0

(x - 5)(x + 2) = 0

x-5 = 0 or x+2 = 0

x = 5 or x = -2

If x = -2, then AH = x-2 = -2-2 = -4, but having a negative segment length is not possible. Lengths must be positive. So we ignore x = -2

If x = 5, then we find the following lengths

Then note how AH*HG = 3*12 = 36 while FH*HB = 9*4 = 36. This confirms that (AH)*(HG) = (FH)*(HB) is a true equation and confirms we have the correct x value.

complete the steps to find the area of the trapezoid.

area of rectangle = 256   square units

area of triangle 1 = 24  square units

area of triangle 2 = 16  square units

area of triangle 3 =   96 square units

area of trapezoid = 120  square units

step-by-step explanation: