Question

Show that points P(1, -2), Q(5, 2), R(3, -1), S(-1, -5) are the vertices of a parallelogram.

Hint:

The midpoint formula method is generally the quickest way to prove a quadrilateral is a parallelogram. Alternatively, you could use the distance formula to show that opposite sides are equal in length (PQ = RS and QR = SP), but this involves more calculations.

Answer 1

Step 1: Understanding the Concept:
A key property of a parallelogram is that its diagonals bisect each other. This means that the midpoint of one diagonal is the same as the midpoint of the other diagonal. We can use the midpoint formula to verify this property for the given points.

Step 2: Key Formula or Approach:
The midpoint of a line segment joining points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula: \[ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

Step 3: Detailed Explanation:
The vertices of the quadrilateral are P(1, -2), Q(5, 2), R(3, -1), and S(-1, -5). The diagonals are PR and QS.
1. Find the midpoint of diagonal PR:
The coordinates are P(1, -2) and R(3, -1). \[ \text{Midpoint of PR} = \left( \frac{1 + 3}{2}, \frac{-2 + (-1)}{2} \right) \] \[ = \left( \frac{4}{2}, \frac{-3}{2} \right) \] \[ = \left( 2, -1.5 \right) \] 2. Find the midpoint of diagonal QS:
The coordinates are Q(5, 2) and S(-1, -5). \[ \text{Midpoint of QS} = \left( \frac{5 + (-1)}{2}, \frac{2 + (-5)}{2} \right) \] \[ = \left( \frac{4}{2}, \frac{-3}{2} \right) \] \[ = \left( 2, -1.5 \right) \] Since the midpoint of diagonal PR is the same as the midpoint of diagonal QS, the diagonals bisect each other.

Step 4: Final Answer:
Because the diagonals of quadrilateral PQRS bisect each other, the points P, Q, R, and S are the vertices of a parallelogram.