In ΔFGH, FH = 8 ft, FG = 13 ft, and m∠F = 72°. Find m∠G. Round your answer to the nearest tenth.
(1 point)
Responses
26.2º
26.2º
35.9º
35.9º
72.1º
72.1º
32.5º
To find the measure of angle G, we can use the Law of Cosines.
The Law of Cosines states that for a triangle with sides a, b, and c, and angle C across from side c, the following equation holds:
c^2 = a^2 + b^2 - 2ab*cos(C)
In our triangle, we have side a = FG = 13 ft, side b = FH = 8 ft, and angle C = ∠F = 72°. We want to find the measure of angle G, which is angle across from side GH.
Using the Law of Cosines, we have:
GH^2 = FG^2 + FH^2 - 2*FG*FH*cos(72°)
GH^2 = 13^2 + 8^2 - 2*13*8*cos(72°)
GH^2 = 169 + 64 - 208*cos(72°)
GH^2 = 233 - 208*cos(72°)
Using a calculator, we find that cos(72°) ≈ 0.3090.
GH^2 = 233 - 208*0.3090
GH^2 = 233 - 64.472
GH^2 ≈ 168.528
Taking the square root of both sides, we have:
GH ≈ √(168.528)
GH ≈ 12.98 ft (rounded to the nearest hundredth)
Now that we have the length of side GH, we can use the Law of Cosines again to find the measure of angle G:
GH^2 = FH^2 + GH^2 - 2*FH*GH*cos(G)
12.98^2 = 8^2 + 12.98^2 - 2*8*12.98*cos(G)
168.528 = 64 + 168.528 - 207.68*cos(G)
207.68*cos(G) = 64
cos(G) = 64/207.68
cos(G) ≈ 0.3086
Now, we can use the inverse cosine function to find the measure of angle G:
G = cos^(-1)(0.3086)
G ≈ 72.1° (rounded to the nearest tenth)