Question

if ( ef = 84 ), ( fd = 85 ), ( ed = 112 ), ( ig = 68 ), and ( hg = 89.6 ), find the perimeter of ( \triangle ghi ). round your answer to the nearest tenth if necessary. figures are not necessarily drawn to scale.

Explanation:

Step1: Determine similarity of triangles

Since the angles of \(\triangle DEF\) and \(\triangle GHI\) are equal (\(\angle F=\angle I = 83^{\circ}\), \(\angle D=\angle G=48^{\circ}\), \(\angle E=\angle H = 49^{\circ}\)), \(\triangle DEF\sim\triangle GHI\) by the AAA (Angle - Angle - Angle) similarity criterion.

Step2: Find the scale factor

The scale factor \(k\) is given by the ratio of corresponding sides. We know that \(EF = 84\) and \(IG=68\). The scale factor \(k=\frac{IG}{FD}=\frac{68}{85}=0.8\)

Step3: Find the lengths of the other sides of \(\triangle GHI\)

For side \(HI\): Using the scale factor, if \(ED = 112\) (corresponds to \(HG\) in the similar triangle relation, but actually, since \(\triangle DEF\sim\triangle GHI\), \(HI\) corresponds to \(EF\)). \(HI=EF\times k=84\times0.8 = 67.2\)
For side \(GH\): Given \(HG = 89.6\) (already given, but also \(GH\) corresponds to \(ED\). \(ED\times k=112\times0.8=89.6\) (matches the given value)

Step4: Calculate the perimeter of \(\triangle GHI\)

The perimeter \(P\) of \(\triangle GHI\) is \(P=HI + IG+HG\). Substitute \(HI = 67.2\), \(IG = 68\), and \(HG=89.6\)
\(P=67.2+68 + 89.6\)
\(P=224.8\)

Answer:

\(224.8\)