2760.21 – Adjacent Corner of a Square

The area of a square is 81. One corner of the square is at the point (5,−4)(5, -4). Which of these points might be an adjacent corner?

  1. (5−62,1)(5 - 6\sqrt2, 1)

  2. (−8,62 −4)(-8, 6\sqrt2 - 4)

  3. (2,62 −4)(2, 6\sqrt2 - 4)

  4. (62 −5,−1)(6\sqrt2 - 5, -1)

  5. (8,4−62)(8, 4-6\sqrt2)

Solution

The sides of the square are length 9. We could plug each of the candidate points into the distance formula. Or we could observe that the vertex will be on a circle having equation (x −5)2 +(y +4)2 =81(x - 5)^2 + (y + 4)^2 = 81.

Either way, after some computational hassle, we see that (c) works: (2,62 −4)(2, 6\sqrt2 - 4).