Question

The three vertices of a rhombus PQRS are P(2, \(-\)3), Q(6, 5) and R(\(-\)2, 1). Find the coordinates of the fourth vertex S and coordinates of the point where both the diagonals PR and QS intersect.

Hint:

The property of diagonals bisecting each other is true for all parallelograms, including rectangles, rhombuses, and squares.

Answer 1

Step 1: Understanding the Concept:
The diagonals of a rhombus bisect each other. This means they have the same midpoint.
Step 2: Key Formula or Approach:
Midpoint formula: \( M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \).
Step 3: Detailed Explanation:
Let the intersection point be \( M(x, y) \). It is the midpoint of diagonal \( PR \).
Vertices \( P(2, -3) \) and \( R(-2, 1) \).
\[ M = \left(\frac{2 + (-2)}{2}, \frac{-3 + 1}{2}\right) = \left(\frac{0}{2}, \frac{-2}{2}\right) = (0, -1) \]
Now, \( M(0, -1) \) is also the midpoint of diagonal \( QS \).
Let \( S = (x_s, y_s) \). Vertex \( Q(6, 5) \).
Using the midpoint formula for \( QS \):
\[ \frac{6 + x_s}{2} = 0 \implies 6 + x_s = 0 \implies x_s = -6 \]
\[ \frac{5 + y_s}{2} = -1 \implies 5 + y_s = -2 \implies y_s = -7 \]
Step 4: Final Answer:
Coordinates of intersection point are (0, \(-1\)) and the fourth vertex S is (\(-6\), \(-7\)).