Question

iva v: geometry sem a- a
ng triangle similarity theorems
what is the length of \\( \overline{aj} \\) if \\( \overline{ab} \parallel \overline{jk} \\)?
10.25 in.
14 in.
8.75 in.
8 in.
(image of 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 (on the side) is 5.25 in, and point j at the top.)

Explanation:

Step1: Identify Similar Triangles

Since \( \overline{AB} \parallel \overline{JK} \), by the Basic Proportionality Theorem (Thales' theorem), \( \triangle HAB \sim \triangle HJK \). Thus, the ratios of corresponding sides are equal: \( \frac{HA}{HJ} = \frac{HB}{HK} \).

Step2: Determine Lengths of Segments

Given \( HB = 3 \) in, \( BK = 5 \) in, so \( HK = HB + BK = 3 + 5 = 8 \) in. Let \( AJ = x \), then \( HJ = HA + AJ = 5.25 + x \).

Step3: Set Up Proportion and Solve

Substitute into the proportion: \( \frac{5.25}{5.25 + x} = \frac{3}{8} \). Cross - multiply: \( 5.25\times8 = 3\times(5.25 + x) \). Calculate left - hand side: \( 42 = 15.75+3x \). Subtract 15.75 from both sides: \( 42 - 15.75 = 3x \), so \( 26.25 = 3x \). Divide by 3: \( x=\frac{26.25}{3}=8.75 \) in.

Answer:

8.75 in.