prove or disprove that the quadrilateral points Q(-3,4), R(5,2), S(4,-1), and T(-4,1) is a rectangle.

Question

trying to prove and disprove quadrilateral points. Could you please show me how you arrived to your answer thanks.

Answer 1

A rectangle is a closed figure that has two pairs of parallel sides.  The parallel sides are equal and all four angles are 90 degrees.

 

1)

 

The best way to do this is to plot the four points on a coordinate system.  Then connect them, so that you obtain your desired figure. 

 

 

2)

 

Use the distance formula to find the lengths of the sides formed by connecting the points.  Then determine the slope of opposite sides to see if their slopes are the same.  Same slope indicates parallelism.

Answer 2

First, plot the points on graph paper.  By inspection, it appears to be at least a parallelogram.

 

There are several ways to show that it is a rectangle.  here is one:

By inspection, we see that QR is parallel to TS, and TQ is parallel to SR

therefore, it is a rectangle if the triangle QRS is a right triangle

 

Lengths:

Q - R = (-3, 4) - (5, 2) = -8, 2  (magnitude = √(64 + 2)

 

R - S = (5, 2) - (4, -1) = 1, 3 (magnitude = √(1 + 9)

 

S - Q = (4, -1) - (-3, 4) = 7, -5 (magnitude = √(49 + 25)

 

Test for right triangle:

SQ^2 = RS^2 + QR^2 ?

49 + 25 = 1 + 9 + 64 + 2 ?

 

74 ≠ 76, therefore the triangle is not a right triangle, and the figure is not a rectangle.

 

You can also see this by checking the angles with an angle gauge, scrap of paper, etc.