The coordinates of the vertices of quadrilateral JKLM are J(-3,2), K(4,-1), L(2,-5) and M(-5,-2).

Find the slope of each side of the quadrilateral and determine if the quadrilateral is a parallelogram?

Question

The coordinates of the vertices of quadrilateral JKLM are J(-3,2), K(4,-1), L(2,-5) and M(-5,-2).

Find the slope of each side of the quadrilateral and determine if the quadrilateral is a parallelogram?

Caroline Aucoin · Accepted Answer

JKLM is a parallelogram

The slope #m# of a line through two points #(x_1, y_1)# and #(x_2, y_2)# is given by the formula: #m = (Delta y)/(Delta x) = (y_2 - y_1)/(x_2 - x_1)# So the slopes of the sides of our quadrilateral are: #m_(JK) = ((-1)-2)/(4-(-3)) = -3/7# #m_(KL) = ((-5)-(-1))/(2-4) = 2# #m_(LM) = ((-2)-(-5))/(-5-2) = -3/7# #m_(MJ) = (2-(-2))/((-3)-(-5)) = 2# So #JK# is parallel to #LM# and #KL# is parallel to #MJ# So #JKLM# is a parallelogram.

Maverick Smith · Answer

undefined The slopes of the sides of the quadrilateral JKLM can be calculated using the formula:

$$ \text{Slope} = \frac{y_2 - y_1}{x_2 - x_1} $$

Using the given coordinates:

1. Slope of side JK:
$$ \text{Slope}_{JK} = \frac{-1 - 2}{4 - (-3)} = \frac{-3}{7} $$

2. Slope of side KL:
$$ \text{Slope}_{KL} = \frac{-5 - (-1)}{2 - 4} = \frac{-4}{-2} = 2 $$

3. Slope of side LM:
$$ \text{Slope}_{LM} = \frac{-2 - (-5)}{-5 - 2} = \frac{3}{-7} = -\frac{3}{7} $$

4. Slope of side MJ (which is the same as the slope of side KL):
$$ \text{Slope}_{MJ} = 2 $$

A quadrilateral is a parallelogram if opposite sides are parallel. For a quadrilateral to be a parallelogram, the slopes of opposite sides must be equal.

Here, the slopes are:
- Side JK and side LM: $-\frac{3}{7} \neq -\frac{3}{7}$
- Side KL and side MJ: $2 = 2$

Since only one pair of opposite sides have equal slopes, the quadrilateral JKLM is not a parallelogram.