In parallelogram ABCD , diagonals AC¯¯¯¯¯ and BD¯¯¯¯¯ intersect at point E, AE=x2−16 , and CE=6x . What is AC ? Enter your answer in the box. units

Question

This is for some geometry work im doing and i got lost here

Answer 1

The diagonals of a parallelogram bisect each other. That means AE = EC. Based on the algebraic expressions given to represent AE and EC the following equation must be true:

x² - 16 = 6x

x² - 6x - 16 = 0

x² - 8x + 2x - 16 =0

x(x-8) + 2(x-8) = 0

(x+2)(x-8) = 0

x = -2 or x = 8

When x = -2, 6x = -12 which cannot be the length of segment EC.

When x = 8, 8² - 16 = 64 - 16 = 48 AND 6(8) = 48. This confirms that diagonal AC has a midpoint at point E.

And since E is between A and C, AE + EC = AC. 48 + 48 = 96.