Respuesta :

Given:

The three vertices of parallelogram DEFG are D(-4, - 2), E(-3, 1) and F(3, 3).

To find:

The coordinates of vertex G.

Solution:

Let the coordinates of vertex G are (a,b).

We know that, diagonal of parallelogram bisect equal other. It means, the midpoints of both diagonals are same.

Midpoint formula:

[tex]Midpoint=\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right)[/tex]

Two diagonals of the parallelogram DEFG are DF and EG.

[tex]\text{Midpoint of DF}=\text{Midpoint of EG}[/tex]

[tex]\left(\dfrac{-4+3}{2},\dfrac{-2+3}{2}\right)=\left(\dfrac{-3+a}{2},\dfrac{1+b}{2}\right)[/tex]

[tex]\left(\dfrac{-1}{2},\dfrac{1}{2}\right)=\left(\dfrac{-3+a}{2},\dfrac{1+b}{2}\right)[/tex]

On comparing both sides, we get

[tex]\dfrac{-3+a}{2}=-\dfrac{1}{2}[/tex]

[tex]-3+a=-1[/tex]

[tex]a=-1+3[/tex]

[tex]a=2[/tex]

And,

[tex]\dfrac{1+b}{2}=\dfrac{1}{2}[/tex]

[tex]1+b=1[/tex]

[tex]b=1-1[/tex]

[tex]b=0[/tex]

Therefore, the coordinates of vertex G are (2,0).

akposevictor

The coordinates of vertex G of parallelogram DEFG is: (2, 0).

Recall:

  • Diagonals of a parallelogram bisects each other, meaning they divide each other into equal parts.
  • Midpoint formula = [tex](\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})[/tex]

The given vertices of parallelogram DEFG are:

D(-4, - 2), E(-3, 1) and F(3, 3)

Let (x, y) = coordinates of vertex G.

DF and EG are diagonals of parallelogram DEFG, therefore:

  • midpoint of DF = midpoint of EG

Find midpoint of D(-4, - 2) and F(3, 3) using  [tex](\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})[/tex]:

 [tex](\frac{-4 + 3}{2}, \frac{-2 + 3}{2}) = (\frac{-1}{2}, \frac{1}{2})[/tex]

Find midpoint of E(-3, 1) and G(x, y) using  [tex](\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})[/tex]:

 [tex](\frac{-3 + x}{2}, \frac{1 + y}{2})[/tex]

Make the midpoint of DF equal to the midpoint of EG, then solve for x and y.

[tex]\frac{-1}{2}, \frac{1}{2} = \frac{-3 + x}{2}, \frac{1 + y}{2}[/tex]

  • Solve for x:

[tex]\frac{-1}{2} = \frac{-3 + x}{2}[/tex]

  • Cross multiply

[tex]2(-1) = 2(-3 + x)\\\\-2 = -6 + 2x\\\\-2 + 6 = 2x\\\\4 = 2x\\\\\mathbf{x = 2}[/tex]

  • Solve for y:

[tex]\frac{1}{2} = \frac{1 + y}{2}[/tex]

  • Cross multiply

[tex]2(1) = 2(1 + y)\\\\2 = 2 + 2y\\\\2 - 2 = 2y\\\\0 = 2y\\\\\mathbf{y = 0}[/tex]

Therefore, the coordinates of vertex G of parallelogram DEFG is: (2, 0).

Learn more about parallelogram on:

https://brainly.com/question/3050890

Otras preguntas