On the coordinate plane (6, 2) and (0, 6) are the endpoints of the diagonal of a square. What are the coordinates of the

[BTGmoderatorDC]

On the coordinate plane (6, 2) and (0, 6) are the endpoints of the diagonal of a square. What are the coordinates of the point on the corner of the square which is closest to the origin?

(A) (0, 1)
(B) (1, 0)
(C) (1, 1)
(D) (2, 0)
(E) (2, 2)



OA C

Source: Veritas Prep

[Ian Stewart]

If you understand slopes well, you can answer the question without much algebra. Recall that perpendicular slopes are negative reciprocals. So if you travel, say, 5 units to the right and 2 units up along one line, then to travel the same distance along a perpendicular line, you would need to travel 2 units to the left and 5 units up (we reverse one direction because the perpendicular slope is negative, and we exchange the horizontal and vertical movements because the slopes are reciprocals). Using that here:

The diagonals of a square meet at their midpoints. The midpoint of (6, 2) and (0, 6) is (3, 4) (averaging the x and y coordinates). That must also be the midpoint of the other diagonal.

The diagonals of a square are also perpendicular. We know on the diagonal from (0, 6) to (6, 2), to go from (0, 6) to (3, 4), we go right 3 units and down 2 units. To travel the same distance from (3, 4) along the perpendicular diagonal, we would need to go left 2 units and down 3 units. That will take us to the point (1, 1), so that must be a vertex of the square.

[swerve]

On the coordinate plane (6, 2) and (0, 6) are the endpoints of the diagonal of a square. What are the coordinates of the point on the corner of the square which is closest to the origin?

(A) (0, 1)
(B) (1, 0)
(C) (1, 1)
(D) (2, 0)
(E) (2, 2)



OA C

Source: Veritas Prep

Length of the diagonal \(=\sqrt{36+16} = \sqrt{52}\)

\(\sqrt{2}\cdot\)Side \(= \sqrt{52}\)

Length of each side \(=\sqrt{26}\)

Test the length in the given options, we find that only C satisfies the length of the side of the square.

Therefore, the correct answer is C.