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Quantitative Aptitude Question on Coordinate Geometry

Let ABCD be a parallelogram such that the coordinates of its three vertices A, B, C are (1, 1), (3, 4), and (−2, 8), respectively. Then, the coordinates of the vertex D are

A

(−4, 5)

B

(4, 5)

C

(−3, 4)

D

(0, 11)

Answer

(−4, 5)

Explanation

Solution

The correct answer is :A
Since ABCD is a parallelogram, opposite sides are parallel and have the same length. The midpoint formula can be used to find the midpoint of a line segment given its endpoints.
First, let's find the midpoint of AB:
Midpoint of AB=((x1+x2)2,(y1+y2)2)=((1+3)2,(1+4)2)=(2,2.5)(\frac{(x1 + x2)}{2},\frac{(y1 + y2)}{2})=(\frac{(1 + 3)}{2},\frac{(1 + 4)}{2})=(2, 2.5)
Next, let's find the midpoint of BC:
Midpoint of BC=((x1+x2)2,(y1+y2)2)(\frac{(x1 + x2)}{2},\frac{(y1 + y2)}{2})=((3+(−2))2,(4+8)2)(\frac{(3 + (-2))}{2},\frac{(4 + 8)}{2})=(0.5, 6)
Now, since opposite sides of a parallelogram are parallel, the vector from A to B is equal to the vector from D to C. We can use this information to find the coordinates of point D.
Vector from A to B: (3-1, 4-1)=(2,3)
Coordinates of C: (-2, 8)
Now, we can add the vector AB to the coordinates of C to find D:
D = C + AB = (-2 + 2, 8 + 3) = (0, 11)
So, the correct answer is:
The coordinates of the vertex D are (0, 11).