Question

equivalent experience_ma.912.gr.1.1_unit 6
find each measure if g is the incenter of \\(\triangle abc\\)

\\(m\angle gaf = \\)

\\(eg = \\) (round your answer to the nearest hundredth if necessary)

Explanation:

Step1: Find ∠BCA

∠BCA = 2×30° = 60°

Step2: Find ∠BAC

First, calculate sum of known angles: 42° + 60° = 102°
∠BAC = 180° - 102° = 78°

Step3: Calculate ∠GAF

∠GAF = $\frac{1}{2}$×∠BAC = $\frac{1}{2}$×78° = 39°

Step4: Find length of DG

DG = AG×sin(∠GAD) = 7.49×sin(39°)

Step5: Calculate EG

Since DG=EG (incenter distances to sides equal):
EG = 7.49×sin(39°) ≈ 7.49×0.6293 ≈ 4.71

Answer:

$m\angle GAF = 39^\circ$
$EG \approx 4.71$