Question

The points Q, left parenthesis, minus, 4, comma, minus, 5, right parenthesis, comma, R, left parenthesis, 8, comma, minus, 8, right parenthesis, comma, S, left parenthesis, 9, comma, minus, 4, right parenthesisQ(−4,−5),R(8,−8),S(9,−4), and T, left parenthesis, minus, 3, comma, minus, 1, right parenthesisT(−3,−1) form quadrilateral QRST. Plot the points then click the "Graph Quadrilateral" button., On this question page, you will first graph points on a coordinate grid. Following the graph are two buttons, "Graph Quadrilateral" will drawn lines between your points, and "Reset" will remove all of your points. Once you press Graph Quadrilateral, if your points are correct, the answer area will appear.
Click on the graph to plot a point. Click a point to delete it.
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y
Q
R
S
T
Use the dropdown menus and answer blanks below to prove the quadrilateral is a rectangle.
Answer
Attempt 1 out of 2

I will prove that quadrilateral QRST is a rectangle by demonstrating that

Answer 1

I will prove that quadrilateral QRST is a rectangle by demonstrating that opposite sides are parallel (so it is a parallelogram) and that two adjacent sides are perpendicular (so it has a right angle).

Compute slopes:
- slope QR = (−8 − (−5)) / (8 − (−4)) = (−3) / 12 = −1/4
- slope RS = (−4 − (−8)) / (9 − 8) = 4 / 1 = 4
- slope ST = (−1 − (−4)) / (−3 − 9) = 3 / (−12) = −1/4
- slope TQ = (−5 − (−1)) / (−4 − (−3)) = (−4) / (−1) = 4

From these slopes:
- QR and ST both have slope −1/4, so QR ∥ ST.
- RS and TQ both have slope 4, so RS ∥ TQ.

Thus opposite sides are parallel, so QRST is a parallelogram.

Also slope QR · slope RS = (−1/4) · 4 = −1, so QR ⟂ RS. Therefore one angle of the parallelogram is a right angle. A parallelogram with one right angle is a rectangle.

Therefore quadrilateral QRST is a rectangle.