Question

quadrilateral jklm has vertices at j(-7, -2), k(3, 1), l(7, -2), and m(-3, -5). is jklm a parallelogram? justify your answer. yes, \\(\overline{jk}\\) and \\(\overline{lm}\\) are congruent. yes, \\(\overline{jk}\\) is parallel to \\(\overline{lm}\\), and \\(\overline{kl}\\) is parallel to \\(\overline{jm}\\). no, \\(\overline{jk}\\) and \\(\overline{lm}\\) are not parallel. no, diagonals \\(\overline{jl}\\) and \\(\overline{km}\\) are not congruent.

Explanation:

Step1: Recall the definition of a parallelogram

A quadrilateral is a parallelogram if both pairs of opposite sides are parallel (i.e., have the same slope).

Step2: Calculate the slope of \(\overline{JK}\)

The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). For \(J(-7, -2)\) and \(K(3, 1)\), we have:
\(m_{JK} = \frac{1 - (-2)}{3 - (-7)} = \frac{3}{10}\)

Step3: Calculate the slope of \(\overline{LM}\)

For \(L(7, -2)\) and \(M(-3, -5)\):
\(m_{LM} = \frac{-5 - (-2)}{-3 - 7} = \frac{-3}{-10} = \frac{3}{10}\)

Step4: Calculate the slope of \(\overline{KL}\)

For \(K(3, 1)\) and \(L(7, -2)\):
\(m_{KL} = \frac{-2 - 1}{7 - 3} = \frac{-3}{4}\)

Step5: Calculate the slope of \(\overline{JM}\)

For \(J(-7, -2)\) and \(M(-3, -5)\):
\(m_{JM} = \frac{-5 - (-2)}{-3 - (-7)} = \frac{-3}{4}\)

Step6: Analyze the slopes

We see that \(m_{JK} = m_{LM} = \frac{3}{10}\) (so \(\overline{JK} \parallel \overline{LM}\)) and \(m_{KL} = m_{JM} = \frac{-3}{4}\) (so \(\overline{KL} \parallel \overline{JM}\)). Since both pairs of opposite sides are parallel, \(JKLM\) is a parallelogram.

Answer:

Yes, \(\overline{JK}\) is parallel to \(\overline{LM}\), and \(\overline{KL}\) is parallel to \(\overline{JM}\).