Question

if ( ec = 14 ), ( cd = 11 ), and ( fg = 30 ), find the length of ( overline{hf} ). round your answer to the nearest tenth if necessary. figures are not necessarily drawn to scale.
answer attempt 1 out of 2
( hf=)

Explanation:

Step1: Determine similarity of triangles

Since the angles of triangle \(ECD\) (\(47^{\circ},58^{\circ},75^{\circ}\)) and triangle \(HFG\) (\(47^{\circ},58^{\circ},75^{\circ}\)) are equal, \(\triangle ECD\sim\triangle HFG\) (by AAA - Angle - Angle - Angle similarity criterion).

Step2: Set up proportion

For similar triangles, the ratios of corresponding sides are equal. That is \(\frac{EC}{HF}=\frac{CD}{FG}\).
Given \(EC = 14\), \(CD = 11\), and \(FG = 30\). Substitute these values into the proportion: \(\frac{14}{HF}=\frac{11}{30}\).

Step3: Solve for \(HF\)

Cross - multiply: \(11\times HF=14\times30\). Then \(HF=\frac{14\times30}{11}\).
Calculate \(14\times30 = 420\), so \(HF=\frac{420}{11}\approx38.2\).

Answer:

\(38.2\)