The diagonals of a rhombus PQRS intersect at T. Given that P (2,2), Q(3,6), R(-1,5):

a. draw the rhombus PQRS on the grid provided
b. State the coordinates of T.

T is the mid-point of segment PR, or

T((2-1)/2,(2+5)/2)=T(1/2,7/2)

To draw the rhombus PQRS on the grid, you need to follow these steps:

a. Plot the given points P(2,2), Q(3,6), and R(-1,5) on the grid.
- P(2,2) is located at the point (2, 2) on the grid.
- Q(3,6) is located at the point (3, 6) on the grid.
- R(-1,5) is located at the point (-1, 5) on the grid.

b. Determine the fourth vertex, S, using the properties of a rhombus.
- Since the diagonals of a rhombus bisect each other at a 90-degree angle, you can find the midpoint of the diagonal PR using the midpoint formula.
- The coordinates of the midpoint of PR = [(x₁ + x₂)/2, (y₁ + y₂)/2]
- The coordinates of the midpoint of PR = [(2 + -1)/2, (2 + 5)/2]
- The coordinates of the midpoint of PR = [1/2, 7/2]

- To find the fourth vertex, S, you need to reflect the midpoint of PR across the midpoint of the other diagonal QS.
- The midpoint of QS can be found using the midpoint formula as well.
- The coordinates of the midpoint of QS = [(x₁ + x₂)/2, (y₁ + y₂)/2]
- The coordinates of the midpoint of QS = [(2 + 3)/2, (2 + 6)/2]
- The coordinates of the midpoint of QS = [5/2, 8/2]
- The coordinates of the midpoint of QS = [5/2, 4]

- Now, reflect the midpoint of PR (1/2, 7/2) across the midpoint of QS (5/2, 4) to find the fourth vertex, S.
- To reflect a point across another point, you can use the formula:
- S = 2Q - R
- Plug in the values:
- S = 2 * (5/2, 4) - (1/2, 7/2)
- S = (5, 8) - (1/2, 7/2)
- S = (5 - 1/2, 8 - 7/2)
- S = (9/2, 9/2)

- The coordinates of S are (9/2, 9/2).

Now that you have the coordinates of all four vertices, you can plot them on the grid and connect them to form the rhombus PQRS.