OQRS IS A QUADRILATERAL SUCH THAT OQ= -6,3 OR= -3,7 AND OS 1,5. T IS THE POINT ON OQ SUCH THAT OT:TQ= 2:2. PROVE QRST IS A PARALLELOGRAM

To prove that QRST is a parallelogram, we need to show that opposite sides are parallel.

Given the coordinates of the points O, Q, R, S, and T, we can find the slope of the line passing through each pair of points and check if they are equal.

Let's find the slopes:

The slope of a line passing through two points (x1, y1) and (x2, y2) is given by the formula:

slope = (y2 - y1) / (x2 - x1)

For the line passing through points OQ(-6, 3) and OR(-3, 7):

slope of OQ = (7 - 3) / (-3 - (-6)) = 4 / 3

For the line passing through points OQ(-6, 3) and TS(1, 5):

slope of TS = (5 - 3) / (1 - (-6)) = 2 / 7

Now, let's find the coordinates of point T on line OQ such that OT:TQ = 2:2.

Since OT:TQ = 2:2, the coordinates of point T will divide the segment OQ into two equal parts.

Using the section formula, the coordinates of point T can be found as follows:

xt = (2 * xq + 2 * xo) / (2 + 2) = (2 * (-6) + 2 * 1) / 4 = (-12 + 2) / 4 = -10 / 4 = -5/2

yt = (2 * yq + 2 * yo) / (2 + 2) = (2 * 3 + 2 * 3) / 4 = (6 + 6) / 4 = 12 / 4 = 3

Therefore, point T is (-5/2, 3).

For the line passing through points QR(-3, 7) and ST(-5/2, 3):

slope of QR = (3 - 7) / (-5/2 - (-3)) = -4 / (5/2 + 6) = -4 / (5/2 + 12/2) = -4 / (17/2) = -8/17

Lastly, let's find the slope of the line passing through points RS(1, 5) and ST(-5/2, 3):

slope of ST = (3 - 5) / (-5/2 - 1) = -2 / (-5/2 - 2/2) = -2 / (-9/2) = 4/9

Thus, we have the following slopes for the lines:

slope of OQ = 4/3
slope of TS = 2/7
slope of QR = -8/17
slope of ST = 4/9

To show that QRST is a parallelogram, we need to prove that opposite sides are parallel.

The opposite sides of a parallelogram have equal slopes. From the slopes we calculated, we can see:

slope of OQ = slope of ST
slope of TS = slope of QR

Therefore, we have proved that QRST is a parallelogram, as opposite sides are parallel.