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

YZ would be negative 4 to a 2

Answer 2

9514 1404 393

Answer:

5

Explanation:

A graph can be helpful. It shows possible integer coordinates of Z could be ...

(-1, 2) or (-5, 1)

These have distances 5 and 4√2 from Y, respectively. Of these, only the length YZ = 5 appears on your list of possible answers.

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The hypotenuse of a right triangle is the diameter of the circumcircle. That is, the right angle vertex (Z) appears on a circle whose center is the midpoint of the hypotenuse. Here, that circle center is C = (-4-1, 2-3)/2 = (-5/2, -1/2). The given point Y differs from this by C - Y = (-5/2 -(-1), -1/2 -(-3)) = (-3/2, 5/2). So, a point with integer coordinates will be 2×(5/2) = 5 units above point Y, which puts it in the 2nd quadrant.

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Adding any of the 8 combinations (±3/2, ±5/2) or (±5/2, ±3/2) to the circle center coordinates will result in integer coordinates for the right angle vertex. Of these, only the ones listed above are in the second quadrant. A couple are on the y-axis, and two of the possibilities are X and Y. The other two are in the third quadrant.