Question

Much appreciated if you could help. :)

Answer 1

Answer:

The quadrilateral is a parallelogram because it has one pair of opposite sides that are both congruent and parallel ⇒ 2nd answer

Step-by-step explanation:

The formula of a distance between two points is $d=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}$

The formula of a slope of a line is $m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

∵ L (-4 , -4) and M (-1 , -5)

∴ $x_{1}$ = -4 and $x_{2}$ = -1

∴ $y_{1}$ = -4 and $y_{2}$ = -5

- Use the formula of the distance to find LM

∵ $LM=\sqrt{(-1--4)^{2}+(-5--4)^{2}}=\sqrt{9+1}$

∴ LM = $\sqrt{10}$

∵ J (1 , 2) and K (-2 , 3)

∴ $x_{1}$ = 1 and $x_{2}$ = -2

∴ $y_{1}$ = 2 and $y_{2}$ = 3

- Use the formula of the distance to find JK

∵ $JK=\sqrt{(-2-1)^{2}+(3-2)^{2}}=\sqrt{9+1}$

∴ JK = $\sqrt{10}$

- ML and JK have equal lengths

∴ LM ≅ JK

Use the formula of the slope to find the slopes of LM and JK

∵ $m_{LM}=\frac{-5--4}{-1--4}=\frac{-5+4}{-1+4}$

∴ $m_{LM}=-\frac{1}{3}$

∵ $m_{JK}=\frac{3-2}{-2-1}=\frac{1}{-3}$

∴ $m_{JK}=-\frac{1}{3}$

- ML and JK have same slopes

∴ LM // JK

∵ LM and JK are opposite sides in the quadrilateral

∵ LM ≅ JK

∵ ML // Jk

- Two opposite sides in the quadrilateral JKLM are congruent

   and parallel

∴ JKLM is a parallelogram

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