In the following figure, ABCD is a parallelogram. AE ⟂ DC and CF ⟂ AD. If AB =…

Solution overview: use coordinates and the distance-from-point-to-line formula.

Step 1: Coordinate setup.

• Place A at (0,0) and put AB on the x-axis, so B = (16,0).

• Let AD = (x,y). Then D = (x,y) and C = B + AD = (16+x, y).

Step 2: Use AE = 8.

DC is parallel to AB and therefore horizontal, so AE is the vertical distance from A to the line y = y. Thus y = AE = 8.

Step 3: Use CF = 10 and the point-to-line distance formula.

• With y = 8, we have C = (16+x,8) and AD = (x,8).

• Distance from C to line AD = |(C_x * AD_y - C_y * AD_x)| / |AD|.

• Compute the numerator: (16 + x)*8 - 8*x = 128.

• So CF = 128 / |AD|. Given CF = 10, we get |AD| = 128 / 10 = 12.8 cm.

Answer: AD = 12.8 cm.