Question

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 \( n \).

Hint:

Using the midpoint of diagonals is the standard and fastest way to find missing coordinates of a parallelogram.

Answer 1

Step 1: Understanding the Concept:
In a parallelogram, the diagonals bisect each other. Therefore, the midpoint of diagonal AC is the same as the midpoint of diagonal BD.
Step 2: Key Formula or Approach:
Midpoint formula: \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \).
Step 3: Detailed Explanation:
Midpoint of AC:
\[ \text{Midpoint}_{AC} = \left( \frac{4 + 4}{2}, \frac{5 + 3}{2} \right) = (4, 4) \]
Midpoint of BD:
\[ \text{Midpoint}_{BD} = \left( \frac{m + 1}{2}, \frac{6 + n}{2} \right) \]
Since the midpoints must coincide:
1. \( \frac{m + 1}{2} = 4 \implies m + 1 = 8 \implies m = 7 \)
2. \( \frac{6 + n}{2} = 4 \implies 6 + n = 8 \implies n = 2 \)
Wait, looking at the coordinates again: A(4,5), B(m,6), C(4,3), D(1,n).
Midpoint of AC: \( x = (4+4)/2 = 4 \), \( y = (5+3)/2 = 4 \). Correct.
Midpoint of BD: \( x = (m+1)/2 = 4 \implies m = 7 \).
\( y = (6+n)/2 = 4 \implies n = 2 \).
Step 4: Final Answer:
The values are \( m = 7 \) and \( n = 2 \).