Question

If three vertices of a rhombus taken in order are (2, -1), (3, 4) and (-2, 3), then the fourth vertex is :

• A. (-3, -2)
• B. (3, 2)
• C. (2, 3)
• D. (1, 2)

Answer 1

The Correct Option is A

To find the fourth vertex of a rhombus given three vertices, we use the property that the diagonals of a rhombus bisect each other at right angles and are equal in length from center to vertex pairs. The given points are A(2,-1), B(3,4), and C(-2,3).

Firstly, we calculate the midpoint of the diagonal that can be defined by segment AC:

Midpoint of AC = ((2 + (-2))/2, (-1 + 3)/2) = (0/2, 2/2) = (0,1).

This midpoint must also be the midpoint of the diagonal BD. Suppose D is the fourth vertex. Let D be (x,y). Then the midpoint of BD is:

Midpoint of BD = ((3 + x)/2, (4 + y)/2).

Since these midpoints are the same, we equate them:

(3 + x)/2 = 0 and (4 + y)/2 = 1.

From (3 + x)/2 = 0, we solve for x:

x = -3.

From (4 + y)/2 = 1, we solve for y:

4 + y = 2.

y = -2.

Thus, the fourth vertex D is (-3,-2), which is the correct option.