Question

if fd = 90, de = 85, and gh = 17, find the length of \\(\overline{ig}\\). round your answer to the nearest tenth if necessary. figures are not necessarily drawn to scale.

Explanation:

Step1: Check triangle similarity

In $\triangle FDE$ and $\triangle IHG$, $\angle F = \angle I = 62^\circ$, $\angle D = 50^\circ$, so $\angle E = 180 - 62 - 50 = 68^\circ$, and $\angle H = 68^\circ$. Thus, $\triangle FDE \sim \triangle IHG$ (AA similarity).

Step2: Set up proportion

Let $IG = x$. The sides opposite equal angles: $DE$ corresponds to $HG$, $FD$ corresponds to $IG$. So $\frac{DE}{HG}=\frac{FD}{IG}$. Substitute values: $\frac{85}{17}=\frac{90}{x}$.

Step3: Solve for x

Cross - multiply: $85x = 17\times90$. Then $x=\frac{17\times90}{85}$. Simplify: $17\times90 = 1530$, $1530\div85 = 18$.

Answer:

18