Question

what is the length of \\(\overline{aj}\\) if \\(\overline{ab} \parallel \overline{jk}\\)?
\\(\circ\\) 8 in.
\\(\circ\\) 8.75 in.
\\(\circ\\) 10.25 in.
\\(\circ\\) 14 in.
(image shows a triangle with points h, b, k on the base, h to b is 3 in, b to k is 5 in, h to a is 5.25 in, and a, j are vertices with ab parallel to jk)

Explanation:

Step1: Identify similar triangles

Since $\overline{AB} \parallel \overline{JK}$, $\triangle HAB \sim \triangle HJK$ (AA similarity).

Step2: Set up proportion

$\frac{HA}{HJ} = \frac{HB}{HK}$. Let $AJ = x$, so $HJ = HA + AJ = 5.25 + x$. $HB = 3$, $HK = 3 + 5 = 8$.

Step3: Solve for x

$\frac{5.25}{5.25 + x} = \frac{3}{8}$
Cross multiply: $5.25 \times 8 = 3(5.25 + x)$
$42 = 15.75 + 3x$
$3x = 42 - 15.75 = 26.25$
$x = \frac{26.25}{3} = 8.75$

Answer:

B. 8.75 in.