Question

The points S, left parenthesis, minus, 2, comma, 1, right parenthesis, comma, T, left parenthesis, minus, 8, comma, minus, 1, right parenthesis, comma, U, left parenthesis, minus, 7, comma, minus, 4, right parenthesisS(−2,1),T(−8,−1),U(−7,−4), and V, left parenthesis, minus, 1, comma, minus, 2, right parenthesisV(−1,−2) form quadrilateral STUV. 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
S
T
U
V
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 STUV is a rectangle by demonstrating that

Answer 1

I will prove that quadrilateral STUV is a rectangle by demonstrating that opposite sides are parallel and one pair of adjacent sides are perpendicular (so the parallelogram has a right angle).

Compute slopes:
- slope ST = (−1 − 1)/(−8 − (−2)) = (−2)/(−6) = 1/3
- slope TU = (−4 − (−1))/(−7 − (−8)) = (−3)/1 = −3
- slope UV = (−2 − (−4))/(−1 − (−7)) = 2/6 = 1/3
- slope VS = (1 − (−2))/(−2 − (−1)) = 3/(−1) = −3

Conclusions from slopes:
- ST and UV have equal slopes (1/3), so ST ∥ UV.
- TU and VS have equal slopes (−3), so TU ∥ VS.
- slope ST · slope TU = (1/3)·(−3) = −1, so ST ⟂ TU (they are perpendicular).

Since both pairs of opposite sides are parallel, STUV is a parallelogram; because one interior angle (between ST and TU) is a right angle, the parallelogram is a rectangle. Therefore STUV is a rectangle.