Probelm Solving - 17

On the coordinate plane (6, 2) and (0, 6) are the endpoints of the diagonal of a square. What is the distance between point (0, 0) and the closest vertex of the square?

(A) 1/sqrt (2)
(B) 1
(C) sqrt (2)
(D) sqrt (3)
(E) 2*sqrt (3)

Answer : C

Midpoint of the diagonal = (3,4)
Slope of the diagonal = -2/3
Slope of the other diagonal is the negative reciprocal of the first diagonal = 3/2

Let one of the vertices = (a,b).

The midpoint (diagonal) and the slope (diagonal) gives us the 1st equation = (y-4)/(x-3) = 3/2

The length of the diagonal = 2*sqrt13.

Calculate the length of the diagonal from the midpoint to the vertex (1/2 of diagonal) .

This gives us our 2nd equation = (x-3)^2 + (y-4)^2 = 13

Solving both the equations :

(x,y) = (1,1)

Thus, the distance is sqrt 2