Question

find j.

(image of a triangle with points g, h, i, k, j. gh is 8 yd, hi is 8 yd, hk is 6 yd, and ji is j. angles at h and i are right angles.)

write your answer as a whole number or a decimal. do not round.

j = \boxed{} yards

Explanation:

Step1: Identify Similar Triangles

Triangles \( \triangle GHK \) and \( \triangle GIJ \) are similar (by AA similarity, as both are right - angled and share \( \angle G \)).

Step2: Set Up Proportion

For similar triangles, the ratios of corresponding sides are equal. Let the height of \( \triangle GHK \) be \( h_1 = 6\) yd, base \( b_1 = 8\) yd, height of \( \triangle GIJ \) be \( j\) (which we need to find), and base \( b_2=16\) yd. The proportion is \( \frac{h_1}{j}=\frac{b_1}{b_2}\), substituting the values we get \( \frac{6}{j}=\frac{8}{16}\).

Step3: Solve for \( j \)

Cross - multiply: \( 8j=6\times16\). Then \( 8j = 96\). Divide both sides by 8: \( j=\frac{96}{8}=12\).

Answer:

12