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:

For any parallelogram (including rectangles, rhombuses, and squares), the sum of opposite vertices' coordinates is equal:
\( x_P + x_R = x_Q + x_S \implies 2 + (-2) = 6 + x \implies x = -6 \).
\( y_P + y_R = y_Q + y_S \implies -3 + 1 = 5 + y \implies y = -7 \).
This vector addition shortcut finds the fourth vertex in just one step!

Answer 1

Step 1: Understanding the Question:
We are given three consecutive vertices of a rhombus \(PQRS\): \(P(2, -3)\), \(Q(6, 5)\), and \(R(-2, 1)\).
We need to find:
1. The coordinates of the fourth vertex \(S(x, y)\).
2. The coordinates of the intersection point of the diagonals.

Step 2: Key Formula or Approach:
1. A rhombus is a special type of parallelogram.
2. In any parallelogram, the diagonals bisect each other. This means the midpoint of diagonal \(PR\) is identical to the midpoint of diagonal \(QS\).
3. The midpoint coordinates of a segment joining \((x_1, y_1)\) and \((x_2, y_2)\) are:
\[ M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]

Step 3: Detailed Explanation:
1. Let the fourth vertex be \(S(x, y)\). Let \(M\) be the intersection point of the diagonals \(PR\) and \(QS\).
2. Since \(M\) is the midpoint of diagonal \(PR\), calculate its coordinates using \(P(2, -3)\) and \(R(-2, 1)\):
\[ M = \left(\frac{2 + (-2)}{2}, \frac{-3 + 1}{2}\right) \]
\[ M = \left(\frac{0}{2}, \frac{-2}{2}\right) = (0, -1) \]
3. This gives the coordinates of the diagonal intersection point directly: \((0, -1)\).
4. Since \(M(0, -1)\) is also the midpoint of diagonal \(QS\), relate it to the coordinates of \(Q(6, 5)\) and \(S(x, y)\):
\[ \left(\frac{6 + x}{2}, \frac{5 + y}{2}\right) = (0, -1) \] 5. Equate the corresponding coordinates to solve for \(x\) and \(y\):
For the \(x\)-coordinate:
\[ \frac{6 + x}{2} = 0 \implies 6 + x = 0 \implies x = -6 \]
For the \(y\)-coordinate:
\[ \frac{5 + y}{2} = -1 \implies 5 + y = -2 \implies y = -7 \]
6. The coordinates of the fourth vertex \(S\) are \((-6, -7)\).

Step 4: Final Answer:
The coordinates of the fourth vertex \(S\) are \((-6, -7)\), and the diagonals intersect at \((0, -1)\).