In ΔFGH, FH = 8 ft, FG = 13 ft, and m∠F = 72°. Find m∠G. Round your answer to the nearest tenth.
A. 26.2º
B. 35.9º
C. 72.1º
D. 32.5º
To find the measure of angle G, we can use the Law of Cosines. Let's call the angle we want to find x.
Using the Law of Cosines:
FH^2 = FG^2 + GH^2 - 2(FG)(GH)cos(x)
Substitute the given values:
8^2 = 13^2 + GH^2 - 2(13)(GH)cos(72°)
64 = 169 + GH^2 - 26GHcos(72°)
Rearranging the equation:
26GHcos(72°) = GH^2 - 105
GH^2 - 26GHcos(72°) - 105 = 0
Using the Law of Cosines:
cos(x) = (FG^2 + GH^2 - FH^2) / 2(FG)(GH)
cos(x) = (13^2 + GH^2 - 8^2) / 2(13)(GH)
cos(x) = (169 + GH^2 - 64) / 26(GH)
cos(x) = (GH^2 - 105) / 26(GH)
Now we have two equations, we can substitute GH^2 - 26GHcos(72°) - 105 for GH^2 in the second equation:
cos(x) = (GH^2 - 105) / 26(GH)
cos(x) = (26GHcos(72°)) / 26(GH)
cos(x) = cos(72°)
x = 72°
Therefore, the measure of angle G is 72°. So, the answer is C. 72.1°.