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.
Show 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!
Step 1: Use the opposite-sides property of a parallelogram instead of the diagonal-midpoint property.
In a parallelogram $ABCD$, the side $AB$ is parallel to and equal in length to the side $DC$, and it points in the same direction. In vector form, this means:
\[ \vec{AB} = \vec{DC} \]
Step 2: Write both vectors in terms of coordinates.
With $A(4,5)$, $B(m,6)$, $C(4,3)$, $D(1,n)$:
\[ \vec{AB} = (m - 4,\ 6 - 5) = (m-4,\ 1) \]
\[ \vec{DC} = (4 - 1,\ 3 - n) = (3,\ 3-n) \]
Step 3: Equate the corresponding components.
Since $\vec{AB} = \vec{DC}$, their $x$-components must match, and their $y$-components must match:
\[ m - 4 = 3 \]
\[ 1 = 3 - n \]
Step 4: Solve each equation.
From the first equation:
\[ m = 3 + 4 = 7 \]
From the second equation:
\[ n = 3 - 1 = 2 \]
Step 5: Final Answer.
The values are $m = 7$ and $n = 2$, so option (A) is correct.
\[ \boxed{m = 7,\ n = 2} \]
