Question

In a right triangle, the hypotenuse has endpoints XY, shown on the graph.

On a coordinate plane, line X Y has points (negative 4, 2), (negative 1, negative 3).

If Z represents the third vertex in the triangle and is located in the second quadrant with integer coordinates, what is the length of YZ?
3
4
5
6

Answer 1

Let X = (-4,2) and Y = (-1,-3). Since XY is the hypotenuse, Z must make a right angle at Z, so ZA · ZB = 0 where A = X, B = Y. For Z = (x,y):

(-4 - x)(-1 - x) + (2 - y)(-3 - y) = 0
=> x^2 +5x + y^2 + y -2 = 0.

Testing integer points in quadrant II (x<0, y>0) gives Z = (-1,2) (note Z = (-4,2) is X itself so not allowed). Then
YZ = distance from (-1,-3) to (-1,2) = |2 - (-3)| = 5.

So the length YZ is 5.