Question

quadrilateral abcd has coordinates a (3, 1), b (4, 4), c (7, 5), d (6, 2). quadrilateral abcd is a
rectangle, because opposite sides are congruent and adjacent sides are perpendicular

square, because all four sides are congruent and adjacent sides are perpendicular

parallelogram, because opposite sides are congruent and adjacent sides are not perpendicular

rhombus, because all four sides are congruent and adjacent sides are not perpendicular

Explanation:

Step1: Calculate side lengths

Use distance formula $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$:

• $AB=\sqrt{(4-3)^2+(4-1)^2}=\sqrt{1+9}=\sqrt{10}$
• $BC=\sqrt{(7-4)^2+(5-4)^2}=\sqrt{9+1}=\sqrt{10}$
• $CD=\sqrt{(6-7)^2+(2-5)^2}=\sqrt{1+9}=\sqrt{10}$
• $DA=\sqrt{(3-6)^2+(1-2)^2}=\sqrt{9+1}=\sqrt{10}$

Step2: Calculate side slopes

Use slope formula $m=\frac{y_2-y_1}{x_2-x_1}$:

• Slope of $AB$: $m_{AB}=\frac{4-1}{4-3}=3$
• Slope of $BC$: $m_{BC}=\frac{5-4}{7-4}=\frac{1}{3}$

Step3: Check perpendicularity

Perpendicular lines have slopes with product $-1$.
$m_{AB} \times m_{BC}=3 \times \frac{1}{3}=1
eq -1$, so adjacent sides are not perpendicular.

Answer:

D. rhombus, because all four sides are congruent and adjacent sides are not perpendicular