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:

For a parallelogram, diagonals bisect each other. For a rectangle, adjacent sides must be perpendicular (check slopes).

Answer 1

Given points:
\(A(6, 1), B(p, 2), C(9, 4), D(7, q)\)
ABCD is a parallelogram.

Step 1: Use the property of parallelogram
In a parallelogram, the diagonals bisect each other.
So, midpoint of \(AC =\) midpoint of \(BD\).

Coordinates of midpoint of \(AC\):
\[ \left( \frac{6 + 9}{2}, \frac{1 + 4}{2} \right) = (7.5, 2.5) \]
Coordinates of midpoint of \(BD\):
\[ \left( \frac{p + 7}{2}, \frac{2 + q}{2} \right) \]

Equate midpoints:
\[ \frac{p + 7}{2} = 7.5 \quad \Rightarrow \quad p + 7 = 15 \quad \Rightarrow \quad p = 8 \]
\[ \frac{2 + q}{2} = 2.5 \quad \Rightarrow \quad 2 + q = 5 \quad \Rightarrow \quad q = 3 \]

Step 2: Check whether ABCD is a rectangle
A parallelogram is a rectangle if adjacent sides are perpendicular.
Check slopes of \(AB\) and \(BC\). If their product is \(-1\), then angle between them is \(90^\circ\).

Calculate slope of \(AB\):
\[ m_{AB} = \frac{2 - 1}{8 - 6} = \frac{1}{2} = 0.5 \]
Calculate slope of \(BC\):
\[ m_{BC} = \frac{4 - 2}{9 - 8} = \frac{2}{1} = 2 \]
Check product:
\[ m_{AB} \times m_{BC} = 0.5 \times 2 = 1 \neq -1 \]
Since product of slopes is not \(-1\), sides \(AB\) and \(BC\) are not perpendicular.
Hence, ABCD is not a rectangle.

Final Answer:
\(p = 8\), \(q = 3\)
ABCD is not a rectangle.