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?

Answer 1

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.
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Answer 2

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

Slope=y2y1x2x1\text{Slope} = \frac{y_2 - y_1}{x_2 - x_1}

Using the given coordinates:

  1. Slope of side JK: SlopeJK=124(3)=37\text{Slope}_{JK} = \frac{-1 - 2}{4 - (-3)} = \frac{-3}{7}

  2. Slope of side KL: SlopeKL=5(1)24=42=2\text{Slope}_{KL} = \frac{-5 - (-1)}{2 - 4} = \frac{-4}{-2} = 2

  3. Slope of side LM: SlopeLM=2(5)52=37=37\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): SlopeMJ=2\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: 3737-\frac{3}{7} \neq -\frac{3}{7}
  • Side KL and side MJ: 2=22 = 2

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

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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