Question

the vertices of quadrilateral lmno are shown.
l (10, 3) m (3, -2) n (-7, 0) o (0, 5)
how is quadrilateral lmno best described?
□ rectangle
□ rhombus
□ parallelogram but not a rectangle or rhombus
□ trapezoid but not a parallelogram, rectangle or rhombus

Explanation:

Step1: Calculate side slopes

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

• $m_{LM}=\frac{-2-3}{3-10}=\frac{-5}{-7}=\frac{5}{7}$
• $m_{MN}=\frac{0-(-2)}{-7-3}=\frac{2}{-10}=-\frac{1}{5}$
• $m_{NO}=\frac{5-0}{0-(-7)}=\frac{5}{7}$
• $m_{OL}=\frac{3-5}{10-0}=\frac{-2}{10}=-\frac{1}{5}$

Step2: Check parallel sides

$m_{LM}=m_{NO}=\frac{5}{7}$, so $LM \parallel NO$
$m_{MN}=m_{OL}=-\frac{1}{5}$, so $MN \parallel OL$
Thus, LMNO is a parallelogram.

Step3: Check right angles

Multiply slopes of adjacent sides:
$m_{LM} \times m_{MN}=\frac{5}{7} \times (-\frac{1}{5})=-\frac{1}{7}
eq -1$
So no right angles, not a rectangle.

Step4: Calculate side lengths

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

• $LM=\sqrt{(3-10)^2+(-2-3)^2}=\sqrt{49+25}=\sqrt{74}$
• $MN=\sqrt{(-7-3)^2+(0-(-2))^2}=\sqrt{100+4}=\sqrt{104}$

$LM
eq MN$, so not a rhombus.

Answer:

Parallelogram but not a rectangle or rhombus