To determine this, we must find the distance between x and y, x and z, and finally between y and z.
The distance formula is
D = sqrt[(y2 - y1)² + (x2 - x1)²]
The distance between x and y is equal to
sqrt[(-2 + 4)² + (2 + 6)²] = sqrt (2² + 8²) = sqrt (4 + 64) = sqrt 68 = sqrt 4 * sqrt 17 = 2 sqrt 17 = 8.2
Distance between x and z is equal to
sqrt[(5 + 4)² + (0 + 6)²] = sqrt(9² + 6²)] = sqrt(81 + 36) = sqrt 117 = sqrt 9 * sqrt 13 = 3 sqrt 13 = 10.8
Finally, the distance between y and z is equal to
sqrt[(5 + 2)² + (0 - 2)²] = sqrt (7² + (-2)²) = sqrt (49 + 4) = sqrt 53 = 7.3
Since our middle calculation is the largest value, that would have to be length of the hypotenuse if these three points do indeed form a right triangle. To determine this, we can use the Pythagorean Theorem, which says that
a² + b² = c² , where c is the hypotenuse and a and b are the legs of the right triangle
Since we rounded to the nearest tenth to determine which value was the largest, I will use the radical form when applying the theorem
Our legs would be 2 sqrt 17 and sqrt 53
(2 sqrt 17)² + (sqrt 53)² = 4(17) + 53 = 68 + 53 = 121 = c²
11 = c (of course we disregard the negative root since dimensions cannot be negative)
However, our hypotenuse would have to be equal to 3 sqrt 13 in order for this to be a right triangle
11 is not equal to 3 sqrt 13, so this is not a right triangle
