+129850 First one
Law of Cosines
(x + 4)^2 = x^2 + (x + 2)^2 - 2(x * (x + 2))cos 120° (cos 120° = -1/2)
x^2 + 8x + 16 = x^2 + x^2 + 4x + 4 + x (x + 2)
x^2 + 8x + 16 = 2x^2 + 4x + 4 + x^2 + 2x
x^2 + 8x + 16 = 3x^2 + 6x + 4
2x^2 - 2x - 12 = 0
x^2 - x - 6 = 0
(x - 3) ( x + 2) = 0
x = 3
+129850 Second one
Let N be the center of the circle and its radius = r
Let M be the tangent point of the circle to OA
OM = 3
ON = 5 - r
MN = r
We can form a right triangle ONM such that
OM^2 + MN^2 = ON^2
3^2 + r^2 = (5-r)^2
3^2 + r^2 = r^2 - 10r + 25
10r = 25 - 9
10r = 16
r = 16 / 10 = 8/5 = 1.6
