Help with Geometry Problems?

Jul 2010
4
0
I'm totally stumped on how to do these:

1. Let P and Q be the points on the sides AB and BC of a tringle ABC respectively such that BP = 3PA and QC = 2BQ. Let K be the midpoint of the segment PQ. Prove that the area of the triangle AKC is equal to \(\displaystyle 11S/24\), where S is the area of the triangle ABC.

2. Let ABC be a triangle such that angle ACB = 135°. Prove that:

\(\displaystyle AB^2=AC^2+BC^2+\sqrt{2}\timesAC\timesBC\)

Thanks
 
Dec 2007
3,184
558
Ottawa, Canada
1. Let P and Q be the points on the sides AB and BC of a tringle ABC respectively such that BP = 3PA and QC = 2BQ. Let K be the midpoint of the segment PQ. Prove that the area of the triangle AKC is equal to \(\displaystyle 11S/24\), where S is the area of the triangle ABC.
TRY it using right triangle 3-4-5:
Code:
 C

(2)

 Q
(1)
 B     (3)      P(1)A
Place point K on QP as per the problem.
Area ABC = 6 (easy part!)
You'll get area AKC = 11/4
11(6) / 24 = 11/4
So that worked! I'm too lazy to try "any" triangle (Sleepy)