1. **State the problem:** We have an isosceles right triangle XYZ with points Z(2, 3) and X(5, 3). The right angle is at X. We need to find the coordinates of vertex Y. 2. **Understand the properties:** In an isosceles right triangle, the two legs are equal in length and meet at the right angle. Since the right angle is at X, the legs are ZX and XY. 3. **Calculate length of leg ZX:** $$ZX = \sqrt{(5-2)^2 + (3-3)^2} = \sqrt{3^2 + 0^2} = 3$$ 4. **Since triangle is isosceles right, length XY = ZX = 3.** 5. **Find coordinates of Y:** Since ZX is horizontal (both points have y=3), XY must be vertical (perpendicular to ZX) with length 3. 6. **Coordinates of X are (5,3), so Y is either 3 units up or down:** - Up: (5, 3 + 3) = (5, 6) - Down: (5, 3 - 3) = (5, 0) 7. **Check which point forms a right triangle with Z and X:** - Vector ZX = (5-2, 3-3) = (3, 0) - Vector XY up = (5-5, 6-3) = (0, 3) - Vector XY down = (5-5, 0-3) = (0, -3) 8. **Dot product of ZX and XY must be zero for right angle at X:** - For Y up: $3 \times 0 + 0 \times 3 = 0$ - For Y down: $3 \times 0 + 0 \times (-3) = 0$ Both satisfy the right angle condition. 9. **Conclusion:** Both (5, 6) and (5, 0) are possible coordinates for Y. Usually, the vertex above the base is chosen, so **Final answer:** $$Y = (5, 6)$$