Question

choose the most specific classification of the following quadrilateral created by vertices a(7,4), b(2, -1), c(-3,4), and d(2,9) parallelogram rectangle rhombus scalene quadrilateral square

Explanation:

Step1: Calculate side lengths

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

• $AB=\sqrt{(2-7)^2+(-1-4)^2}=\sqrt{25+25}=\sqrt{50}=5\sqrt{2}$
• $BC=\sqrt{(-3-2)^2+(4-(-1))^2}=\sqrt{25+25}=\sqrt{50}=5\sqrt{2}$
• $CD=\sqrt{(2-(-3))^2+(9-4)^2}=\sqrt{25+25}=\sqrt{50}=5\sqrt{2}$
• $DA=\sqrt{(7-2)^2+(4-9)^2}=\sqrt{25+25}=\sqrt{50}=5\sqrt{2}$

Step2: Calculate diagonal lengths

• $AC=\sqrt{(-3-7)^2+(4-4)^2}=\sqrt{100+0}=10$
• $BD=\sqrt{(2-2)^2+(9-(-1))^2}=\sqrt{0+100}=10$

Step3: Check angle properties

Verify if sides are perpendicular using slope formula $m=\frac{y_2-y_1}{x_2-x_1}$:

• Slope of $AB$: $m_{AB}=\frac{-1-4}{2-7}=1$
• Slope of $BC$: $m_{BC}=\frac{4-(-1)}{-3-2}=-1$
• Product: $m_{AB} \times m_{BC}=1 \times (-1)=-1$, so $AB \perp BC$

Answer:

square