If the points A(4, 5), B(m, 6), C(4, 3) and D(1, n) taken in this order are the vertices of a parallelogram ABCD, then find the values of m and n.
Generated by claude-sonnet-4-6 · 2026-06-15 10:30 · grounding rag
Model Answer
In a parallelogram, the diagonals bisect each other.
So, mid-point of AC = mid-point of BD.
$$\left(\frac{4+4}{2},\ \frac{5+3}{2}\right) = \left(\frac{m+1}{2},\ \frac{6+n}{2}\right)$$
$$\left(4,\ 4\right) = \left(\frac{m+1}{2},\ \frac{6+n}{2}\right)$$
Comparing x-coordinates: $\dfrac{m+1}{2} = 4 \Rightarrow m = 7$
Comparing y-coordinates: $\dfrac{6+n}{2} = 4 \Rightarrow n = 2$
Therefore, m = 7 and n = 2.
Source: Chapter 7, Section 7.3 (Mid-point formula / Example 10)
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Explanation
- The key property used is: diagonals of a parallelogram bisect each other, so mid-point of diagonal AC = mid-point of diagonal BD.
- Apply the mid-point formula $\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$ to both diagonals and equate x- and y-coordinates separately.
- This is directly modelled on Example 10 of the textbook. Examiners award 1 mark for setting up the mid-point equation correctly and 1 mark for the correct values of m and n.