Question

May I please get help with this. I have tried multiple times but still could not get the correct answer

Answer 1

a) length of JK and GH = √58

b) slope of JK and GH = -7/3

c) The quadrilateral is a parallelogram because it has one pair of opposite sides that are both congruent and parallel (option B)

Step-by-step explanation:

Vertices of GHJK:

G(4, -4), H(1, 3), J (-4, 2) and K (-1, -5)

To determine the length of JK and GH, we will apply the distance formula:

`dis\tan ce\text{ = }\sqrt[]{(y_2-y_1)^2+(x_2-x_1)^2}`
`\begin{gathered} \text{length JK = distance betwen J and K} \\ J(-4,2)\text{ }and\text{ }K(-1,-5) \\ \text{distance JK = }\sqrt[]{(-5-2)^2+(-1-(-4))^2} \\ \text{distance JK = }\sqrt[]{(-7)^2+(-1+4)^2}\text{ = }\sqrt[]{49\text{ + }}9 \\ \text{distance JK = }\sqrt[]{58} \end{gathered}`
`\begin{gathered} \text{length GH = distance betwen J and K} \\ G\mleft(4,-4\mright)\text{ }and\text{ }H\mleft(1,3\mright) \\ \text{distance GH = }\sqrt[]{(1-4)^2+(3-(-4))^2} \\ \text{distance GH = }\sqrt[]{(-3)^2+(3+4)^2}=\text{ }\sqrt[]{9+49} \\ \text{distance GH = }\sqrt[]{58} \end{gathered}`

This shows length JK and GH are equal and opposite to each other, Hence, the pair is opposite and congruent.

b) To find the slope of JK and GH, we use the slope formula:

`\begin{gathered} m\text{ = }(y_2-y_1)/(x_2-x_1) \\ \text{for JK: }x_1=-4,y_1=2,x_2=-1,y_2\text{ = -}5 \\ \text{slope = }(-5-2)/(-1-(-4)) \\ \text{slope = }(-7)/(3) \end{gathered}`
`\begin{gathered} x_1=4,y_1=-4,x_2=1,y_2\text{ = 3} \\ \text{slope = }(3-(-4))/(1-4) \\ \text{slope = }(7)/(-3)\text{ } \\ \text{slope =}\frac{\text{-7}}{3} \end{gathered}`

This shows the slopes of JK and GH are equal.

For two lines to be parallel, their slopes must be equal. Since the slopes of JK and GH are equal, we can say they are parallel to each other.

c) A parallelogram is a quadrilateral that has two pairs of parallel sides and opposite sides are equal.

But if one pair is congrruent and parallel, it an also satisfy a paralleloram.

The quadrilateral is a parallelogram because it has one pair of opposite sides that are both congruent and parallel.