Question

Determine whether the quadrilateral is a parallelogram using the indicated method. D(-8,1), E(-3, 6), F(7, 4), G(2, -1)

(DISTANCE FORMULA) the answer should be yes or no. please show the work of it

Answer 1

We can use the distance formula to determine whether the quadrilateral DEFG is a parallelogram. If the opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.The distance formula is:d = sqrt((x2 - x1)^2 + (y2 - y1)^2)Using the distance formula, we can find the lengths of the four sides of the quadrilateral:DE = sqrt((-3 - (-8))^2 + (6 - 1)^2) = sqrt(25 + 25) = sqrt(50)

EF = sqrt((7 - (-3))^2 + (4 - 6)^2) = sqrt(100 + 4) = sqrt(104)

FG = sqrt((2 - 7)^2 + (-1 - 4)^2) = sqrt(25 + 25) = sqrt(50)

GD = sqrt((-8 - 2)^2 + (1 - (-1))^2) = sqrt(100 + 4) = sqrt(104)Since DE = FG and EF = GD, opposite sides are congruent. Therefore, the quadrilateral DEFG is a parallelogram.Answer: Yes

Explanation:

Answer 2

Yes, quadrilateral DEFG is a parallelogram.

Explanation:

The opposite sides of a parallelogram are congruent.

To determine if quadrilateral DEFG is a parallelogram, substitute each pair of adjacent vertices into the distance formula to find the length of each side.

`\boxed{\begin{minipage}{7.4 cm}\underline{Distance between two points}\\\\$d=√((x_2-x_1)^2+(y_2-y_1)^2)$\\\\\\where $(x_1,y_1)$ and $(x_2,y_2)$ are the two points.\\\end{minipage}}`

`\begin{aligned}DE&=√((x_E-x_D)^2+(y_E-y_D)^2)\\&=√((-3-(-8))^2+(6-1)^2)\\&=√((5)^2+(5)^2)\\&=√(25+25)\\&=√(50)\\&=√(25 \cdot 2)\\&=√(25) √(2)\\&=5√(2)\end{aligned}`

`\begin{aligned}EF&=√((x_F-x_E)^2+(y_F-y_E)^2)\\&=√((10)^2+(-2)^2)\\&=√(100+4)\\&=√(104)\\&=√(4 \cdot 26)\\&=√(4) √(26)\\&=2√(26)\\\end{aligned}`

`\begin{aligned}FG&=√((x_G-x_F)^2+(y_G-y_F)^2)\\&=√((-5)^2+(-5)^2)\\&=√(25+25)\\&=√(50)\\&=√(25 \cdot 2)\\&=√(25) √(2)\\&=5√(2)\end{aligned}`

`\begin{aligned}GD&=√((x_D-x_G)^2+(y_D-y_G)^2)\\&=√((-10)^2+(2)^2)\\&=√(100+4)\\&=√(104)\\&=√(4 \cdot 26)\\&=√(4) √(26)\\&=2√(26)\\\end{aligned}`

The opposite sides of quadrilateral DEFG are:

• DE and FG
• EF and GD

As DE = FG = 5√2 units and EF = GD = 2√(26), this proves that quadrilateral DEFG is a parallelogram since its opposite sides are congruent.