Question

using triangle similarity theorems
what is the length of \\(\overline{aj}\\) if \\(\overline{ab} \parallel \overline{jk}\\)?
8 in.
8.75 in.
10.25 in.
14 in.
(image of triangle with points h, b, k on a base, h to b is 3 in, b to k is 5 in, h to a (on hypotenuse) is 5.25 in, and a connected to b, j at the top of the triangle)

Explanation:

Step1: Identify Similar Triangles

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

Step2: Define Variables

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

Step3: Set Up Proportion

Substitute into the proportion: \( \frac{5.25}{5.25 + x} = \frac{3}{8} \).

Step4: Cross-Multiply and Solve

Cross-multiplying: \( 5.25 \times 8 = 3 \times (5.25 + x) \)
\( 42 = 15.75 + 3x \)
Subtract 15.75: \( 42 - 15.75 = 3x \)
\( 26.25 = 3x \)
Divide by 3: \( x = \frac{26.25}{3} = 8.75 \)

Answer:

8.75 in.