Question

in parallelogram abcd, ae = x² - 8 and ce = 2x. what is ac? 4 8 16 24

Explanation:

Step1: Recall parallelogram diagonals property

In a parallelogram, diagonals bisect each other. So, \( AE = CE \).
Given \( AE = x^2 - 8 \) and \( CE = 2x \), we set them equal: \( x^2 - 8 = 2x \).

Step2: Solve quadratic equation

Rearrange the equation: \( x^2 - 2x - 8 = 0 \).
Factor the quadratic: \( (x - 4)(x + 2) = 0 \).
This gives solutions \( x = 4 \) or \( x = -2 \). Since length can't be negative, we take \( x = 4 \).

Step3: Find \( AE \) and \( CE \)

Substitute \( x = 4 \) into \( CE = 2x \), so \( CE = 2(4) = 8 \). Since \( AE = CE \), \( AE = 8 \).

Step4: Calculate \( AC \)

In a parallelogram, \( AC = AE + CE \). So, \( AC = 8 + 8 = 16 \).

Answer:

16