Square ABCD has area of 320 square units. Points K, L, M, N are midpoints of the sides of ABCD, P is the point of intersection of KC and BN, and Q is the point of intersection of AM and BN. Find PQ

Answer 1

Use parallel lines to show that length of BP = length of PQ

Angle ABN = arctan (1/2)

The side of the square is 8 sqrt(5)

Therefore BP= 4 sqrt(5) times cos angle ABN.

I will leave the numerical calculations to you.

OK, I found an easier way!

Triangle ABQ is a right triangle in which the square of the hypotenuse is 320

and the ratio of the legs is 2:1; therefore, (2x)²+x²=320.

x=8

Since the length of the longer leg BQ is twice the length of the shorter leg AQ,

length of PQ = x=8.