A(-1,0),B(3,1) and C(2,2) are the vertices of a parallelogram ABCD. The coordinates of the fourth
vertex D is
Answers
Step-by-step explanation:
Given:
The points A(-1,0), B(3,1), C(2,2)
To find:
The fourth vertex D of the parallelogram ABCD
Calculation:
Let the fourth vertex is D(x,y). In a parallelogram the diagonals bisect each other i,e., Midpoints of the diagonals are equal
In the parallelogram ABCD, AC and BD are two diaognals. Hence their midpoints are equal
\Rightarrow Midpoint\ of \ AC \ = Midpoint \ of \ BD⇒Midpoint of AC =Midpoint of BD
\begin{gathered}\Rightarrow (\frac{-1+2}{2}, \frac{0+2}{2})= (\frac{3+x}{2}, \frac{1+y}{2})\\\\\Rightarrow (\frac{1}{2}, \frac{2}{2})= (\frac{3+x}{2}, \frac{1+y}{2})\\\\\end{gathered}
⇒(
2
−1+2
,
2
0+2
)=(
2
3+x
,
2
1+y
)
⇒(
2
1
,
2
2
)=(
2
3+x
,
2
1+y
)
By comparing x and y co-ordinates on both sides, we get
\begin{gathered}\Rightarrow \frac{3+x}{2}= \frac{1}{2}\ \ \ ,\ \ \ \frac{1+y}{2}= \frac{2}{2}\\\\\Rightarrow3+x=1\ \ \ ,\ \ \ 1+y=2\\\\\Rightarrow x=-2\ \ \ ,\ \ \ y=1\\\\\end{gathered}
⇒
2
3+x
=
2
1
,
2
1+y
=
2
2
⇒3+x=1 , 1+y=2
⇒x=−2 , y=1
\boxed{The\ fourth \ vertex \ is \ D(-2,1)}
The fourth vertex is D(−2,1)
