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.
Generated by claude-sonnet-4-6 · 2026-06-15 10:31 · grounding rag
Model Answer
Finding p and q:
In a parallelogram, diagonals bisect each other, so mid-point of AC = mid-point of BD.
Mid-point of AC = $\left(\dfrac{6+9}{2},\ \dfrac{1+4}{2}\right) = \left(\dfrac{15}{2},\ \dfrac{5}{2}\right)$
Mid-point of BD = $\left(\dfrac{p+7}{2},\ \dfrac{2+q}{2}\right)$
Equating:
$$\frac{p+7}{2} = \frac{15}{2} \Rightarrow p = 8$$
$$\frac{2+q}{2} = \frac{5}{2} \Rightarrow q = 3$$
Checking for rectangle:
For ABCD to be a rectangle, adjacent sides must be equal or diagonals must be equal.
$AC = \sqrt{(9-6)^2+(4-1)^2} = \sqrt{9+9} = 3\sqrt{2}$
$BD = \sqrt{(7-8)^2+(3-2)^2} = \sqrt{1+1} = \sqrt{2}$
Since $AC \neq BD$, ABCD is not a rectangle.
Source: Chapter 7, Section 7.3 (Mid-point Formula)
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Explanation
- The key property used is: diagonals of a parallelogram bisect each other, so their mid-points are equal. Set up two equations (one for x, one for y) to find p and q.
- For checking rectangle: a parallelogram is a rectangle if and only if its diagonals are equal. Calculate both diagonals using the distance formula and compare.
- Examiners award marks for: (1) correct mid-point setup, (2) values of p and q, (3) distance calculation and conclusion. Don't skip the conclusion statement.