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 standard and fastest way to find missing coordinates of a parallelogram.

Answer 1

Given:
A(4, 5), B(m, 6), C(4, 3), D(1, n) are the vertices of parallelogram ABCD in order.

In a parallelogram, the diagonals bisect each other.
So midpoint of AC = midpoint of BD.

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Step 1: Midpoint of AC
A(4,5), C(4,3)

Midpoint of AC =
\[ \left( \frac{4+4}{2},\ \frac{5+3}{2} \right) = (4,\ 4) \]

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Step 2: Midpoint of BD
B(m,6), D(1,n)

Midpoint of BD =
\[ \left( \frac{m+1}{2},\ \frac{6+n}{2} \right) \]

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Step 3: Equate the midpoints
Compare x-coordinates:
\[ \frac{m+1}{2} = 4 \] Multiply by 2:
\[ m + 1 = 8 \] \[ m = 7 \]

Compare y-coordinates:
\[ \frac{6+n}{2} = 4 \] Multiply by 2:
\[ 6 + n = 8 \] \[ n = 2 \]

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Final Answers:
\[ \boxed{m = 7,\ n = 2} \]