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: Use the fact that opposite sides of a parallelogram are equal and parallel, expressed as vectors, instead of the diagonal midpoint property.
In parallelogram $ABCD$ taken in order, side $AB$ is equal and parallel to side $DC$. In vector form this means:
\[ \vec{AB} = \vec{DC} \]

Step 2: Write both vectors using the given coordinates.
$A(4, 5)$, $B(m, 6)$, $C(4, 3)$, $D(1, n)$.
\[ \vec{AB} = (m - 4,\ 6 - 5) = (m - 4,\ 1) \]
\[ \vec{DC} = (4 - 1,\ 3 - n) = (3,\ 3 - n) \]

Step 3: Equate the corresponding components, since equal vectors must match in both the $x$ and $y$ directions.
\[ m - 4 = 3 \quad \Rightarrow \quad m = 7 \]
\[ 1 = 3 - n \quad \Rightarrow \quad n = 3 - 1 = 2 \]

Step 4: State the values found.
$m = 7$ and $n = 2$.

Final Answer:
The values are $m = 7$ and $n = 2$, which corresponds to option (A). \[ \boxed{m = 7,\ n = 2} \]