Question

b is the mid - point of $overline{ac}$, d is the mid - point of $overline{ce}$, $bd = 3x$, and $ae = 60$. find the length of ab if $ac = 5x + 10$.

Explanation:

Step1: Use mid - point property

Since B is the mid - point of $\overline{AC}$ and D is the mid - point of $\overline{CE}$, by the mid - point theorem for line segments, $BD=\frac{1}{2}AE$. Given $BD = 3x$ and $AE=60$, we have $3x=\frac{1}{2}\times60$.
$3x = 30$

Step2: Solve for x

Divide both sides of the equation $3x = 30$ by 3.
$x=\frac{30}{3}=10$

Step3: Find the length of AC

Given $AC = 5x+10$, substitute $x = 10$ into the equation.
$AC=5\times10 + 10=50 + 10=60$

Step4: Find the length of AB

Since B is the mid - point of $\overline{AC}$, $AB=\frac{1}{2}AC$.
$AB=\frac{1}{2}\times60 = 30$

Answer:

30