Question

identify the figure with the vertices q(-3,2), r(1,2), s(1,-4), and t(-3,-4). select choice

Explanation:

Step1: Analyze coordinates of Q and R

Points \( Q(-3,2) \) and \( R(1,2) \) have the same y - coordinate. The distance between them is \( |1 - (-3)|=\sqrt{(1 + 3)^2+(2 - 2)^2}=4 \) (using distance formula \( d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} \), here \( y_2 - y_1 = 0 \)).

Step2: Analyze coordinates of R and S

Points \( R(1,2) \) and \( S(1,-4) \) have the same x - coordinate. The distance between them is \( |-4 - 2|=\sqrt{(1 - 1)^2+(-4 - 2)^2}=6 \) (using distance formula, here \( x_2 - x_1 = 0 \)).

Step3: Analyze coordinates of S and T

Points \( S(1,-4) \) and \( T(-3,-4) \) have the same y - coordinate. The distance between them is \( |-3 - 1|=\sqrt{(-3 - 1)^2+(-4+4)^2}=4 \) (using distance formula, here \( y_2 - y_1 = 0 \)).

Step4: Analyze coordinates of T and Q

Points \( T(-3,-4) \) and \( Q(-3,2) \) have the same x - coordinate. The distance between them is \( |2-(-4)|=\sqrt{(-3 + 3)^2+(2 + 4)^2}=6 \) (using distance formula, here \( x_2 - x_1 = 0 \)).

Step5: Check angles (slopes)

Slope of \( QR \): \( m_{QR}=\frac{2 - 2}{1+3}=0 \) (horizontal line)
Slope of \( RS \): \( m_{RS}=\frac{-4 - 2}{1 - 1} \) (undefined, vertical line)
Since the product of slope of \( QR \) (0) and slope of \( RS \) (undefined) implies perpendicular lines (horizontal and vertical lines are perpendicular). Similarly, other adjacent sides are also perpendicular. And opposite sides are equal ( \( QR = ST = 4 \), \( RS = TQ = 6 \) )

Answer:

The figure is a rectangle (since it has four right angles and opposite sides equal)