Posted by **Anonymous** on Saturday, March 2, 2013 at 2:02pm.

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 -
**Steve**, Saturday, March 2, 2013 at 7:38pm
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

## Answer This Question

## Related Questions

- geometry - I have triangle ABC. Bottom is AC with midpoint F. There is a point ...
- geometry - Triangle ABC, AB=8, BC=10, and AC=12, M is the midpoint of AB, and N ...
- Please help in triangle - ABC is an isosceles triangle with AB=BC and ∠ABC...
- geometry - In the accompanying diagram of triangle ABC side DE is parallel to AC...
- Geometry - Is it possible for AB = 10, BC = 2x + 5 and AC = 5x, given that point...
- geometry - in triangle GHJ, K(2,3) is the midpoint of GH, L(4,1) is the midpoint...
- mats - in a triangle ABC, D is the midpoint of BC and E is the midpoint of AD. ...
- Maths - In a triangle ABC, D is the midpoint of BC and E is the midpoint of AD...
- geometry - In triangle DEF, A(2,3) is the midpoint of DE, B(4,1) is the midpoint...
- geometry - in triangle GHJ, k(2,3) is the midpoint of GH, l(4,1) is the midpoint...

More Related Questions