Question

If the points $A(6, 1)$, $B(p, 2)$, $C(9, 4)$ and $D(7, q)$ are the vertices of a parallelogram $ABCD$, then find the values of $p$ and $q$. Hence, check whether $ABCD$ is a rectangle or not.

Hint:

Use midpoint formula to solve parallelogram diagonal problems; use dot product to test right angles.

Answer 1

Given:
In a parallelogram, the diagonals bisect each other.
We are given the coordinates:
- A(6, 1), C(9, 4)
- B(p, 2), D(7, q)

Step 1: Use midpoint property of diagonals
Since diagonals bisect each other, the midpoint of diagonal AC must be equal to the midpoint of diagonal BD.

Step 2: Find midpoint of AC
\[ \text{Midpoint of } AC = \left( \frac{6 + 9}{2}, \frac{1 + 4}{2} \right) = \left( \frac{15}{2}, \frac{5}{2} \right) = (7.5, 2.5) \]
Step 3: Find midpoint of BD
\[ \text{Midpoint of } BD = \left( \frac{p + 7}{2}, \frac{2 + q}{2} \right) \]
Step 4: Equate the midpoints
\[ \frac{p + 7}{2} = 7.5 \Rightarrow p + 7 = 15 \Rightarrow p = 8 \] \[ \frac{2 + q}{2} = 2.5 \Rightarrow 2 + q = 5 \Rightarrow q = 3 \]
So, \( \boxed{p = 8}, \quad \boxed{q = 3} \)

Step 5: Check if ABCD is a rectangle
To check if the parallelogram is a rectangle, we need to verify if adjacent sides are perpendicular.
First, calculate vectors:
- \( \vec{AB} = B - A = (8 - 6, 2 - 1) = (2, 1) \)
- \( \vec{BC} = C - B = (9 - 8, 4 - 2) = (1, 2) \)

Compute dot product:
\[ \vec{AB} \cdot \vec{BC} = (2)(1) + (1)(2) = 2 + 2 = 4 \]
Since dot product ≠ 0, the angle between AB and BC is not \(90^\circ\).
Therefore, the parallelogram is not a rectangle.

Final Answer:
Values: \( \boxed{p = 8, \ q = 3} \)
Nature of figure: \( \boxed{ABCD \text{ is a parallelogram but not a rectangle}} \)