Question

Points A(-2,4) , B(1,3), C(4,1) and D form a parallelogram.What are the coordinates of D?

Answer 1

Answer:

The coordinates of vertex D are (1,2).

Step-by-step explanation:

Let the coordinates of D are (a,b).

It is given that ABCD is a parallelogram and the vertices of the parallelogram are A(-2,4), B(1,3) and C(4,1).

According to the property of parallelogram, the diagonals of the parallelogram bisect each other.

AC and BD are diagonals of the parallelogram ABCD.

Midpoint formula:

$Midpoint=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$

Using midpoint formula, the midpoint of AC is

$Midpoint_{AC}=(\frac{-2+4}{2},\frac{4+1}{2})=(1,\frac{5}{2})$

Using midpoint formula, the midpoint of BD is

$Midpoint_{BD}=(\frac{1+a}{2},\frac{3+b}{2})$

Midpoint of both diagonals is the intersection point of the diagonals.

$Midpoint_{AC}=Midpoint_{BD}$

$(1,\frac{5}{2})=(\frac{1+a}{2},\frac{3+b}{2})$

On comparing both the sides we get

$1=\frac{1+a}{2}$

$2=1+a$

$2-1=a$

$1=a$

The value of a is 1.

$\frac{5}{2}=\frac{3+b}{2}$

$5=3+b$

$5-3=b$

$2=b$

The value of b is 2.

Therefore the coordinates of vertex D are (1,2).