Question

in the diagram below, gd = 3.8, he = 3, and fg = 6.3. find the length of \\(\overline{fh}\\). round your answer to the nearest tenth if necessary.

answer attempt 1 out of 2
fh =

Explanation:

Step1: Identify Similar Triangles

From the diagram, \( \triangle FGH \sim \triangle FDE \) (by AA similarity, as corresponding angles are equal). So, the ratios of corresponding sides are equal: \( \frac{FG}{FD}=\frac{FH}{FE} \). Also, \( FD = FG + GD = 6.3 + 3.8 = 10.1 \), and \( FE = FH + HE = FH + 3 \).

Step2: Set Up Proportion

Using the similarity ratio: \( \frac{FG}{FD}=\frac{FH}{FE} \) becomes \( \frac{6.3}{10.1}=\frac{FH}{FH + 3} \).

Step3: Cross - Multiply and Solve

Cross - multiply: \( 6.3(FH + 3)=10.1FH \).
Expand: \( 6.3FH+18.9 = 10.1FH \).
Subtract \( 6.3FH \) from both sides: \( 18.9 = 10.1FH - 6.3FH \).
Simplify: \( 18.9 = 3.8FH \).
Solve for \( FH \): \( FH=\frac{18.9}{3.8}\approx4.9737\approx5.0 \) (rounded to the nearest tenth).

Answer:

\( 5.0 \)