If (1, 2), (4, y), (x, 6) and (3, 5) are the vertices of a parallelogram taken in order, find x and y.
Step-by-Step Solution
Key Concept: In a parallelogram the diagonals bisect each other. Hence the mid‑point of one diagonal equals the mid‑point of the other diagonal.
Let the vertices be taken in order as \(A(1,2),\; B(4,y),\; C(x,6),\; D(3,5)\).
1. Use the property of diagonals: In a parallelogram, the mid‑point of diagonal \(AC\) is the same as the mid‑point of diagonal \(BD\).
2. Find the mid‑point of \(AC\):
$$\text{Midpoint of } AC = \left(\frac{1+x}{2},\; \frac{2+6}{2}\right) = \left(\frac{1+x}{2},\; 4\right).$$
3. Find the mid‑point of \(BD\):
$$\text{Midpoint of } BD = \left(\frac{4+3}{2},\; \frac{y+5}{2}\right) = \left(\frac{7}{2},\; \frac{y+5}{2}\right).$$
4. Equate the two mid‑points (since they must be identical):
\[
\frac{1+x}{2} = \frac{7}{2} \quad \text{and} \quad 4 = \frac{y+5}{2}.
\]
5. Solve for \(x\):
\[
\frac{1+x}{2} = \frac{7}{2} \Rightarrow 1 + x = 7 \Rightarrow x = 6.
\]
6. Solve for \(y\):
\[
4 = \frac{y+5}{2} \Rightarrow y + 5 = 8 \Rightarrow y = 3.
\]
7. Verification (optional): Check that opposite sides are parallel and equal.
- \(AB = (4-1,\; y-2) = (3,1)\) and \(CD = (3-6,\; 5-6) = (-3,-1)\) → same magnitude, opposite direction.
- \(BC = (x-4,\; 6-y) = (2,3)\) and \(AD = (3-1,\; 5-2) = (2,3)\) → equal.
Hence the quadrilateral is indeed a parallelogram.
Therefore, \(x = 6\) and \(y = 3\).
Correct Answer: x = 6, \; y = 3