Question

in a right triangle, the hypotenuse has endpoints xy, shown on the graph. 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?

Explanation:

Step1: Determine coordinates of X and Y

From the graph, \(X(-4,2)\) and \(Y(-1,-3)\).

Step2: Analyze right - triangle properties

Since \(Z\) is in the second quadrant (where \(x<0,y > 0\)) and has integer coordinates. For a right - triangle with hypotenuse \(XY\), if we consider vertical and horizontal segments.
If we assume the right - angle is at \(Z\). The horizontal distance from \(Y\) to \(Z\) (change in \(x\)): Let \(Z\) have the same \(y\) - coordinate as \(X\) (to form a right - angle). If \(Z\) has \(y = 2\) (same as \(X\)'s \(y\) - coordinate).
The vertical distance from \(Y(-1,-3)\) to \(Z(x,2)\) (using the distance formula for vertical distance \(d=\vert y_2 - y_1\vert\)).
The length of \(YZ\) (vertical distance when \(Z\) has \(x=-1\) (to be in the second quadrant with integer coordinates)). Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), when \(x_1=-1,y_1=-3,x_2=-1,y_2 = 2\), we have \(YZ=\vert2-(-3)\vert=5\)

Answer:

5