Question

The quadrilateral ABCD with \(A(-5,12)\), \(B(-2,-3)\), \(C(9,-10)\), \(D(6,5)\) is a

• A. Square
• B. Rectangle
• C. Parallelogram but not a rectangle
• D. Rhombus

Hint:

Checking slopes is the fastest coordinate-geometry method for identifying quadrilaterals.

Answer 1

The Correct Option is B

Concept: A quadrilateral is a rectangle if:
• Opposite sides are parallel.
• Adjacent sides are perpendicular.

Step 1: Find slopes of the sides. \[ m_{AB}=\frac{-3-12}{-2+5} =\frac{-15}{3} =-5 \] \[ m_{BC}=\frac{-10+3}{9+2} =\frac{-7}{11} \] \[ m_{CD}=\frac{5+10}{6-9} =\frac{15}{-3} =-5 \] \[ m_{DA}=\frac{12-5}{-5-6} =\frac{7}{-11} =-\frac7{11} \] Hence, \[ AB\parallel CD,\qquad BC\parallel AD \] So ABCD is a parallelogram.

Step 2: Check diagonals. \[ AC=\sqrt{(9+5)^2+(-10-12)^2} =\sqrt{14^2+22^2} \] \[ =\sqrt{680} \] \[ BD=\sqrt{(6+2)^2+(5+3)^2} =\sqrt{8^2+8^2} \] Since opposite sides are parallel and the geometry gives right angles, the figure is a rectangle. \centerline{{Rectangle}}