Question

**#5.) in the diagram below ab bisects cd. if cd = x^2 - 12 yd. and ce = 2x yd. determine the length of ed.

Explanation:

Step1: Recall the property of bisection

If a line segment $\overline{AB}$ bisects $\overline{CD}$ at point $E$, then $CE = ED$. Also, $CD=CE + ED$ and since $CE = ED$, we have $CD = 2CE$.

Step2: Set up the equation

Given $CD=x^{2}-12$ and $CE = 2x$, and $CD = 2CE$. Substitute the expressions:
\[x^{2}-12=2(2x)\]
\[x^{2}-12 = 4x\]
\[x^{2}-4x - 12=0\]

Step3: Solve the quadratic equation

Factor the quadratic equation $x^{2}-4x - 12=(x - 6)(x+ 2)=0$.
Setting each factor equal to zero gives $x - 6=0$ or $x+2=0$. So $x = 6$ or $x=-2$. But since length cannot be negative in this context, we take $x = 6$.

Step4: Find the length of $ED$

Since $ED=CE$ and $CE = 2x$, when $x = 6$, $ED=2x=12$ yd.

Answer:

$12$ yd