Question

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

Answer 1

96 units

Explanation:

If ABCD is parallelogram, then its diagonals bisect each other. This means that

• AE=EC;
• BE=ED.

Since
`AE=x^2-16` and
`EC=6x,` you get

`x^2-16=6x,\\ \\x^2-6x-16=0,\\ \\D=(-6)^2-4\cdot 1\cdot (-16)=36+64=100,\\ \\x_(1,2)=(-(-6)\pm √(100))/(2\cdot 1)=(6\pm 10)/(2)=8,\ -2.`

The distance cannot be negative, thus,
`x=6\ units.`

`AE=8^2-16=64-16=48,\\ \\CE=6\cdot 8=48,\\ \\AC=AE+EC=48+48=96\ units.`