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 most efficient way to solve for unknown coordinates in a parallelogram. Avoid using the distance formula unless necessary.

Answer 1

Step 1: Understanding the Concept:
In a parallelogram, the diagonals bisect each other. This means 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:
1. Midpoint of \(AC\):
\[ M_{AC} = \left( \frac{4 + 4}{2}, \frac{5 + 3}{2} \right) = \left( \frac{8}{2}, \frac{8}{2} \right) = (4, 4) \]
2. Midpoint of \(BD\):
\[ M_{BD} = \left( \frac{m + 1}{2}, \frac{6 + n}{2} \right) \]
3. Equate the midpoints:
For x-coordinates:
\[ \frac{m + 1}{2} = 4 \Rightarrow m + 1 = 8 \Rightarrow m = 7 \]
For y-coordinates:
\[ \frac{6 + n}{2} = 4 \Rightarrow 6 + n = 8 \Rightarrow n = 2 \]
Step 4: Final Answer:
The values are \(m = 7\) and \(n = 2\).