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

To find the length of YZ in the right triangle with hypotenuse XY, we first need to determine the coordinates of points X and Y.

Given the endpoints of the hypotenuse:
- Point X: \((-4, 2)\)
- Point Y: \((-1, -3)\)

Now let's calculate the length of the hypotenuse XY.

Using the distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\):

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Substituting the coordinates of points X and Y:

\[
d = \sqrt{((-1) - (-4))^2 + ((-3) - 2)^2}
\]
\[
= \sqrt{((-1 + 4))^2 + ((-3 - 2))^2}
\]
\[
= \sqrt{(3)^2 + (-5)^2}
\]
\[
= \sqrt{9 + 25}
\]
\[
= \sqrt{34}
\]

The length of segment XY (the hypotenuse) is \(\sqrt{34}\).

Since Z is the third vertex of the right triangle, and we need to find its position in the second quadrant with integer coordinates, we know that Z must form a right angle with the line segments XZ and YZ.

The slope of line XY is calculated as follows:

\[
\text{slope of XY} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-3 - 2}{-1 - (-4)} = \frac{-5}{3}
\]

Since Z must be the vertex that creates a right angle, it should have a slope that is the negative reciprocal of \(-\frac{5}{3}\), which is \(\frac{3}{5}\) or \(-\frac{3}{5}\). Given that Z is in the second quadrant, we will focus on positions where the line XZ or YZ has a slope of \(-\frac{3}{5}\).

To find integer coordinates for Z, we can start testing positions that also maintain a right angle with the hypotenuse.

Let’s choose a point Z such that it has integer coordinates:

- Coordinates of Z could be \((-5, 1)\), which maintains the slope of \(-3/5\).
- To confirm, let's calculate the distance YZ.

Using distance formula again for point Y \((-1, -3)\) and Z \((-5, 1)\):

\[
d = \sqrt{((-5) - (-1))^2 + (1 - (-3))^2}
\]
\[
= \sqrt{(-5 + 1)^2 + (1 + 3)^2}
\]
\[
= \sqrt{(-4)^2 + (4)^2}
\]
\[
= \sqrt{16 + 16}
\]
\[
= \sqrt{32}
\]

This seems incorrect and may not be necessary based on integer distances.

Let's consider some possible right-angle triangle setups with integer values.

- Let’s try Z at \((-4, -3)\).
Calculating YZ:
\[
d = \sqrt{((-4) - (-1))^2 + ((-3) - (-3))^2}
\]
\[
= \sqrt{(-4 + 1)^2 + 0^2}
\]
\[
= \sqrt{(-3)^2} = 3
\]

The point satisfies being in the second quadrant as \((-4, -3)\) is in the second quadrant. Thus, YZ = 3.

Thus, the length of segment YZ is:

\(\boxed{3}\).