Question

if ( cd = 19 ), ( db = 18 ), ( cb = 20 ), ( ge = 27 ), and ( fe = 30 ), find the perimeter of ( \triangle efg ). round your answer to the nearest tenth if necessary. figures are not necessarily drawn to scale.

Explanation:

Step1: Determine similarity of triangles

Since \(\angle D = \angle G=67^{\circ}\), \(\angle B=\angle E = 59^{\circ}\), \(\angle C=\angle F = 54^{\circ}\), by the AA (Angle - Angle) similarity criterion, \(\triangle DBC\sim\triangle GEF\).

Step2: Find the ratio of similarity

The ratio of similarity \(k\) is given by \(\frac{GE}{DB}\). Given \(DB = 18\) and \(GE=27\), then \(k=\frac{27}{18}=\frac{3}{2}\).

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

If \(CD = 19\), then \(FG=CD\times k=19\times\frac{3}{2}=28.5\). If \(CB = 20\), then \(EF = CB\times k=20\times\frac{3}{2} = 30\) (already given).

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

The perimeter \(P\) of \(\triangle EFG\) is \(P=FE + GE+FG\). Substitute \(FE = 30\), \(GE = 27\), \(FG=28.5\) into the formula. Then \(P=30 + 27+28.5=85.5\).

Answer:

The perimeter of \(\triangle EFG\) is \(85.5\).