Question

Points Band C lies on $\overline{AD}$. Find AC if$\frac{AB}{BD}=\frac{3}{4},\frac{AC}{CD}=\frac{5}{6},$

and$BD=66$ .

Short answer

The length of AC is$52.5$ .

Step-by-step solution

Step 1. Given information.

$\frac{AB}{BD}=\frac{3}{4},\frac{AC}{CD}=\frac{5}{6}$and$BD=66$

Point B and C lie on$\overline{AD}$ .

Step 2. Find the length of AB .

Substitute66 for BD in the equation$\frac{AB}{BD}=\frac{3}{4}$ .

$\begin{array}{c}\frac{AB}{66}=\frac{3}{4}\\ AB=\frac{198}{4}\\ AB=49.5\end{array}$

Step 3. Find the length of the lineAB+BD .

$\begin{array}{c}AB+BD=49.5+66\\ =115.5\end{array}$

From the length if AC and CD5x, let and6x be the measures.

The sum of AC and CD, is equal to the length of line.

$\begin{array}{c}5x+6x=115.5\\ 11x=115.5\\ x=10.5\end{array}$

Step 4. Now, find the length of AC .

Substitute$10.5$ for x in the 5x and 6x.

$\begin{array}{c}AC=5x\\ =5\times 10.5\\ =52.5\end{array}$

Therefore, the length of AC is$52.5$ .