Question

find f.

(image of two right triangles sharing a common angle at f, with triangle hfg having hypotenuse 17 yd, vertical leg 15 yd, and triangle ifj having vertical leg 30 yd, and side f to find)

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

f = □ yards

Explanation:

Step1: Find FG in triangle HFG

In right triangle \( HFG \), using the Pythagorean theorem \( a^2 + b^2 = c^2 \), where \( c = 17 \) yd (hypotenuse) and \( b = 15 \) yd (one leg). Let \( FG = x \), then \( x^2 + 15^2 = 17^2 \). So \( x^2 = 17^2 - 15^2 = 289 - 225 = 64 \), thus \( x = \sqrt{64} = 8 \) yd.

Step2: Identify similar triangles

Triangles \( HFG \) and \( JFI \) are similar (by AA similarity, as both are right triangles and share a vertical angle at \( F \)). So the ratios of corresponding sides are equal. Let \( FI = y \), \( JI = 30 \) yd, \( FG = 8 \) yd, \( HJ \) related to \( f \). The ratio of \( JI \) to \( HG \) is \( \frac{30}{15}=2 \). So the ratio of \( JF \) (which is \( f \)) to \( HF \) (17 yd) should also be 2. Thus \( f = 17\times2 = 34 \) yd.

Answer:

34