Question

Let $P(1,2)$, $Q(a,b)$, $R(5,7)$ and $S(2,3)$ be the vertices of a parallelogram $PQRS$. Then ________.

• A. $a=4, b=2$
• B. $a=6, b=2$
• C. $a=6, b=4$
• D. $a=3, b=2$
• E. $a=4, b=6$

Hint:

For a parallelogram $PQRS$, $P + R = Q + S$ (using coordinate addition).

Answer 1

Answer: (E)

Step 1: Understanding the Concept
A key property of a parallelogram is that its diagonals bisect each other. This means that the midpoint of the diagonal connecting opposite vertices is the same for both diagonals.
Step 2: Key Formula or Approach
The midpoint M of a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:
\[ M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \] For the parallelogram PQRS, the vertices are given in order. The diagonals are PR and QS. Therefore, the midpoint of PR must be equal to the midpoint of QS.
Step 3: Detailed Explanation
1. Find the midpoint of diagonal PR.
The vertices are P(1, 2) and R(5, 7).
Midpoint of PR = \(\left(\frac{1 + 5}{2}, \frac{2 + 7}{2}\right) = \left(\frac{6}{2}, \frac{9}{2}\right) = (3, 4.5)\).
2. Find the midpoint of diagonal QS.
The vertices are Q(a, b) and S(2, 3).
Midpoint of QS = \(\left(\frac{a + 2}{2}, \frac{b + 3}{2}\right)\).
3. Equate the midpoints.
Since the midpoints are the same, we can equate their x-coordinates and y-coordinates.
Equating the x-coordinates:
\[ \frac{a + 2}{2} = 3 \] \[ a + 2 = 6 \] \[ a = 4 \] Equating the y-coordinates:
\[ \frac{b + 3}{2} = 4.5 \] \[ b + 3 = 9 \] \[ b = 6 \] So, the coordinates of Q are (4, 6).
Step 4: Final Answer
The values are a = 4 and b = 6.