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.
x
y
Q
R
S
T
Use the dropdown menus and answer blanks below to prove the quadrilateral is a rectangle.
Answer
Attempt 2 out of 2

I will prove that quadrilateral QRST is a rectangle by demonstrating that
one pair of opposite sides is of equal measure AND parallel
.
To prove one pair of opposite sides are equal:

=

=
The pair of
sides are of
measure.

To prove one pair of opposite sides are parallel:
slope of
=
slope of
=
The slopes of
pair of
sides are
.
That being the case, those sides are
.

Therefore, as a result of these two things taken together, the quadrilateral is a
.

Answer 1

To fill in the blanks and prove QRST is a rectangle:

Compute lengths:
- QR = √[(8 − (−4))^2 + (−8 − (−5))^2] = √(12^2 + (−3)^2) = √153 = 3√17
- ST = √[(-3 − 9)^2 + (−1 − (−4))^2] = √((−12)^2 + 3^2) = √153 = 3√17
- RS = √[(9 − 8)^2 + (−4 − (−8))^2] = √(1 + 16) = √17
- TQ = √[(-4 − (−3))^2 + (−5 − (−1))^2] = √(1 + 16) = √17

So QR = ST and RS = TQ. The pair of opposite sides are of equal measure.

Compute slopes:
- slope QR = (−8 − (−5)) / (8 − (−4)) = (−3)/12 = −1/4
- slope ST = (−1 − (−4)) / (−3 − 9) = 3/(−12) = −1/4
Thus slope QR = slope ST, so QR ∥ ST.

- slope RS = (−4 − (−8)) / (9 − 8) = 4/1 = 4
- slope TQ = (−5 − (−1)) / (−4 − (−3)) = (−4)/(−1) = 4
Thus slope RS = slope TQ, so RS ∥ TQ.

Since adjacent slopes (e.g. slope QR = −1/4 and slope RS = 4) are negative reciprocals, adjacent sides are perpendicular. With opposite sides both equal and parallel (a parallelogram) and one right angle, QRST is a rectangle.