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.

• A. m = 7, n = 2
• B. m = 6, n = 2
• C. m = 7, n = 3
• D. m = 5, n = 4

Hint:

For any parallelogram \(ABCD\) in coordinate order, the sum of the coordinates of opposite vertices is equal:
\[ x_A + x_C = x_B + x_D \implies 4 + 4 = m + 1 \implies m = 7 \]
\[ y_A + y_C = y_B + y_D \implies 5 + 3 = 6 + n \implies n = 2 \]
This summation shortcut is much faster than writing out the full midpoint equations!

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given four coordinates representing the vertices of a parallelogram \(ABCD\) taken in order: \(A(4, 5)\), \(B(m, 6)\), \(C(4, 3)\), and \(D(1, n)\). We need to determine the values of the unknown parameters \(m\) and \(y\)-coordinate \(n\).

Step 2: Key Formula or Approach:
A key geometric property of a parallelogram is that its diagonals bisect each other. This means:
\[ \text{Midpoint of diagonal } AC = \text{Midpoint of diagonal } BD \]
We will use the midpoint formula to set up equations for both coordinates:
\[ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

Step 3: Detailed Explanation:
1. Calculate the coordinates of the midpoint of diagonal \(AC\) joining \(A(4, 5)\) and \(C(4, 3)\):
\[ \text{Midpoint of } AC = \left( \frac{4 + 4}{2}, \frac{5 + 3}{2} \right) = \left( \frac{8}{2}, \frac{8}{2} \right) = (4, 4) \]
2. Calculate the coordinates of the midpoint of diagonal \(BD\) joining \(B(m, 6)\) and \(D(1, n)\):
\[ \text{Midpoint of } BD = \left( \frac{m + 1}{2}, \frac{6 + n}{2} \right) \]
3. Since the diagonals bisect each other, equate the corresponding coordinates of both midpoints:
- Equate the \(x\)-coordinates:
\[ \frac{m + 1}{2} = 4 \]
Multiply both sides by 2:
\[ m + 1 = 8 \implies m = 7 \]
- Equate the \(y\)-coordinates:
\[ \frac{6 + n}{2} = 4 \]
Multiply both sides by 2:
\[ 6 + n = 8 \implies n = 2 \]
This gives \(m = 7\) and \(n = 2\).

Step 4: Final Answer:
The values are \(m = 7\) and \(n = 2\), which corresponds to option (A).