Exercise 41 Page 646

Let's begin with plotting the given points in a coordinate plane. We can connect these two points with a segment.

Since we know that △ XYZ is a right triangle with a right angle at vertex Z, we can deduct that the vertex X is lying on the vertical line x=6.

If △ XYZ is a 45^(∘)-45^(∘)-90^(∘) triangle, then the lengths of segments YZ and ZX are equal, as this triangle is isosceles. Let's evaluate the length of YZ.

Therefore, the third vertex X is 7 units from Z either in a positive or a negative direction. However, as we are asked to find the coordinates of X in Quadrant I, we will consider only the first case.

The coordinates of the vertex X are (6,9).