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:
Another approach involves using vector geometry. We can calculate the midpoints of the diagonals and set them equal.
Step 3: Detailed Explanation:
1. Position Vector of Points:
Let the coordinates of \( A \) be \( (4, 5) \).
Let the coordinates of \( C \) be \( (4, 3) \).
Let the coordinates of \( B \) be \( (m, 6) \).
Let the coordinates of \( D \) be \( (1, n) \).
2. Find the midpoint of diagonal \( AC \):
\( M_{AC} = \left( \frac{4 + 4}{2}, \frac{5 + 3}{2} \right) = (4, 4) \)
3. Find the midpoint of diagonal \( BD \):
\( M_{BD} = \left( \frac{m + 1}{2}, \frac{6 + n}{2} \right) \)
4. Equating the midpoints:
Since the diagonals bisect each other, the midpoint of diagonal \( AC \) is the same as the midpoint of diagonal \( BD \).
For x-coordinates:
\( \frac{m + 1}{2} = 4 \quad \Rightarrow \quad m + 1 = 8 \quad \Rightarrow \quad m = 7 \)
For y-coordinates:
\( \frac{6 + n}{2} = 4 \quad \Rightarrow \quad 6 + n = 8 \quad \Rightarrow \quad n = 2 \)
Step 4: Final Answer:
The values are \( m = 7 \) and \( n = 2 \).