The diagonals of quadrilateral QRST intersect at P (-1,4). QRST has vertices at Q (4,6) and R (-2,5). What must be the coordinates of S and T to ensure that QRST is a parallelogram?

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muzammilbhutia3 muzammilbhutia3 · Answer 1 · 2024-03-10T19:19:10+01:00

Answer:

Since QRST is a parallelogram, the diagonals bisect each other. Thus, the midpoint of the diagonals should be the same.

Let's find the midpoint of QR (diagonal) first:

Midpoint of QR = ((4 - 2)/2, (6 + 5)/2) = (1, 5.5)

Since P is the midpoint of ST (diagonal), we can use the midpoint formula to find the other endpoint:

Midpoint of ST = ((1 + x_S)/2, (5.5 + y_S)/2)

Given that P is (-1, 4), we can set up the equation:

((-1 + x_S)/2, (4 + y_S)/2) = (1, 5.5)

Now, solve for x_S and y_S:

1 + x_S = -2 => x_S = -3

5.5 + y_S = 8 => y_S = 2.5

So, the coordinates of S are (-3, 2.5).

To find T, we know that the diagonals bisect each other. So, the midpoint of QT will also be (1, 5.5).

Let's find T using the midpoint formula:

Midpoint of QT = ((4 + x_T)/2, (6 + y_T)/2)

Given that Q is (4, 6), we can set up the equation:

((4 + x_T)/2, (6 + y_T)/2) = (1, 5.5)

Now, solve for x_T and y_T:

4 + x_T = 2 => x_T = -2

6 + y_T = 11 => y_T = 5

So, the coordinates of T are (-2, 5).