# geometry

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Let ABC be a right triangle with ACB =90, AC =6 , and BC =2. E is the midpoint of AC , and F is the midpoint of AB . If CF and BE intersect at G , then cos(CGB),
in simplest radical form, is ((k square root w)/f) where k , w , and f are positive integers. Find the value of k+w+ f.

• geometry -

Since E and F are midpoints, EF ║ CB, and has length CB/2 = 1
CE = AC/2 = 3

Label angles
CGB = a
EBC = b
FCB = c
so, intriangle CGB, a+b+c = 180 degrees
we want

cos a = cos *180 - (b+c))
= -cos(b+c)
= sinb sinc - cosb cosc

Looking at the right triangles,
EB = √13 and FC = √10, so we have

cos a = 3/√13 * 3/√10 - 2/√13 * 1/√10
= 9/√130 - 2/√130
= 7/√130
= 7√130 / 130
k+w+f = 7+130+130 = 267