Question

proving parallelograms in the coordinate plane
method 1
prove both pairs of opposite sides are congruent.
if
and
, then
abcd is a parallelogram.
use_
method 2
prove both pairs of opposite sides are parallel
if
and
, then
abcd is a parallelogram.
use_
method 3
prove one pair of opposite sides are congruent and parallel
if
and
, then
abcd is a parallelogram.
use_
directions: determine whether the figure is a parallelogram using the distance formula.

1. a(-7, 4), b(1, 2), c(9, -8), d(1, -6)
2. p(-4, 2), q(6, 4), r(11, -2), s(2, -3)

Explanation:

Step1: Recall distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.

Step2: Calculate length of $AB$ for first - set of points

For $A(-7,4)$ and $B(1,2)$:
$AB=\sqrt{(1-(-7))^2+(2 - 4)^2}=\sqrt{(8)^2+(-2)^2}=\sqrt{64 + 4}=\sqrt{68}=2\sqrt{17}$

Step3: Calculate length of $CD$ for first - set of points

For $C(9,-8)$ and $D(1,-6)$:
$CD=\sqrt{(1 - 9)^2+(-6-(-8))^2}=\sqrt{(-8)^2+(2)^2}=\sqrt{64 + 4}=\sqrt{68}=2\sqrt{17}$

Step4: Calculate length of $BC$ for first - set of points

For $B(1,2)$ and $C(9,-8)$:
$BC=\sqrt{(9 - 1)^2+(-8 - 2)^2}=\sqrt{(8)^2+(-10)^2}=\sqrt{64+100}=\sqrt{164}=2\sqrt{41}$

Step5: Calculate length of $DA$ for first - set of points

For $D(1,-6)$ and $A(-7,4)$:
$DA=\sqrt{(-7 - 1)^2+(4-(-6))^2}=\sqrt{(-8)^2+(10)^2}=\sqrt{64 + 100}=\sqrt{164}=2\sqrt{41}$
Since $AB = CD$ and $BC=DA$, the figure $ABCD$ is a parallelogram.

Step6: Calculate lengths for second - set of points

For $P(-4,2)$ and $Q(6,4)$:
$PQ=\sqrt{(6-(-4))^2+(4 - 2)^2}=\sqrt{(10)^2+(2)^2}=\sqrt{100 + 4}=\sqrt{104}=2\sqrt{26}$
For $R(11,-2)$ and $S(2,-3)$:
$RS=\sqrt{(2 - 11)^2+(-3-(-2))^2}=\sqrt{(-9)^2+(-1)^2}=\sqrt{81+1}=\sqrt{82}$
For $Q(6,4)$ and $R(11,-2)$:
$QR=\sqrt{(11 - 6)^2+(-2 - 4)^2}=\sqrt{(5)^2+(-6)^2}=\sqrt{25 + 36}=\sqrt{61}$
For $S(2,-3)$ and $P(-4,2)$:
$SP=\sqrt{(-4 - 2)^2+(2-(-3))^2}=\sqrt{(-6)^2+(5)^2}=\sqrt{36 + 25}=\sqrt{61}$
Since $PQ
eq RS$, the figure $PQRS$ is not a parallelogram.

Answer:

1. The figure $ABCD$ is a parallelogram.
2. The figure $PQRS$ is not a parallelogram.