Question

find the length of $overline{ab}$. 1. $ab=$__________

Explanation:

Step1: Identify similar triangles

Since $BE\parallel CD$, $\triangle ABE\sim\triangle ACD$.

Step2: Set up proportion

The ratios of corresponding sides of similar triangles are equal. So $\frac{AB}{AC}=\frac{AE}{AD}$. Here $AD = 12 + 6=18$, $AE = 12$, and $AC=AB + 5$. Let $AB=x$, then $\frac{x}{x + 5}=\frac{12}{18}$.

Step3: Cross - multiply

Cross - multiplying the proportion $\frac{x}{x + 5}=\frac{12}{18}$ gives $18x=12(x + 5)$.

Step4: Expand and solve for x

Expand the right side: $18x=12x+60$. Subtract $12x$ from both sides: $18x-12x=60$, so $6x = 60$. Divide both sides by 6, we get $x = 10$.

Answer:

$10$