If in a parallelogram ABDC, the coordinates of A, B and C are respectively (1, 2), (3, 4) and (2, 5)

Q: Why does the order $ABDC$ change the answer? Isn't a parallelogram just a parallelogram?

Think of the vertices as points on a path. $ABDC$ means you travel from $A$ to $B$, then to $D$, then to $C$. If you assume $ABCD$, you are traveling from $A$ to $B$ to $C$ to $D$. These are two completely different shapes with different diagonals. In coordinate geometry, the order of vertices is the definition of the polygon's boundary. Respect the order, and the geometry will respect you back!

Q: Is there a faster way to find $D$ without finding the midpoint?

You could use vectors! Since $AD$ and $BC$ are diagonals, the vector sum of opposite vertices is equal: $\vec{A} + \vec{D} = \vec{B} + \vec{C}$. This is essentially the same as the midpoint method but often faster to write. Both methods are perfectly valid and elegant. Choose the one that feels most intuitive to you during the exam.

Q: I got the equation $5x - 3y + 1 = 0$, but the options look slightly different. Did I make a sign error?

Sign errors are the silent killers of JEE scores. Always double-check your cross-multiplication. If you have $3(y - 2) = 5(x - 1)$, it becomes $3y - 6 = 5x - 5$. Moving everything to one side gives $5x - 3y + 1 = 0$. If your signs are flipped, you likely moved terms incorrectly across the equals sign. Take a deep breath and re-expand the brackets slowly.

Question

If in a parallelogram $ABDC$, the coordinates of $A$, $B$ and $C$ are respectively $(1, 2)$, $(3, 4)$ and $(2, 5)$ then the equation of the diagonal $AD$ is:

Options:

(A) $3x + 5y - 13 = 0$

(B) $3x - 5y + 7 = 0$

(C) $5x + 3y - 11 = 0$

(D) $5x - 3y + 1 = 0$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing on a coordinate plane. You are given three points: $A(1, 2)$, $B(3, 4)$, and $C(2, 5)$.

The problem asks for the diagonal $AD$ of a parallelogram named $ABDC$. Stop right there. Did you notice the order? It is $ABDC$, not $ABCD$.

This is a classic JEE trap. If you blindly assume the order is $ABCD$, you will connect the wrong vertices and find the wrong diagonal.

In geometry, the order of vertices defines the path of the perimeter. $A$ connects to $B$, $B$ to $D$, $D$ to $C$, and $C$ back to $A$. This means $AD$ and $BC$ are the diagonals.

### The Geometric Insight

Now that we have the correct order, how do we find vertex $D(a, b)$? We could use slopes, but that is tedious.

Instead, let us use the most powerful property of a parallelogram: its diagonals bisect each other. This means the midpoint of diagonal $AD$ is identical to the midpoint of diagonal $BC$.

This is our logic bridge. It turns a complex coordinate problem into a simple equality.

### The Calculation

Let us find the midpoint $M$ of the diagonal $BC$. Using the midpoint formula:

$$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$$

Plugging in $B(3, 4)$ and $C(2, 5)$, we get:

$$M = \left( \frac{3 + 2}{2}, \frac{4 + 5}{2} \right) = (2.5, 4.5)$$

Now, we apply this to the diagonal $AD$. With $A(1, 2)$ and $D(a, b)$, the midpoint is:

$$\left( \frac{a + 1}{2}, \frac{b + 2}{2} \right)$$

Since the diagonals bisect, we set these equal:

$$\left( \frac{a + 1}{2}, \frac{b + 2}{2} \right) = (2.5, 4.5)$$

Solving for $a$ and $b$ is straightforward:

$$\frac{a + 1}{2} = 2.5 \Rightarrow a + 1 = 5 \Rightarrow a = 4$$

$$\frac{b + 2}{2} = 4.5 \Rightarrow b + 2 = 9 \Rightarrow b = 7$$

We have successfully located vertex $D(4, 7)$.

### The Final Line

We are almost there. We need the equation of the line passing through $A(1, 2)$ and $D(4, 7)$. We use the two-point form:

$$y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)$$

Substituting our points, we get:

$$y - 2 = \frac{7 - 2}{4 - 1}(x - 1)$$

This simplifies to:

$$y - 2 = \frac{5}{3}(x - 1)$$

Cross-multiplying by $3$ gives $3(y - 2) = 5(x - 1)$, or $3y - 6 = 5x - 5$. Rearranging into the standard form $Ax + By + C = 0$, we arrive at the final result:

**$5x - 3y + 1 = 0$**

Animated Solution for Mathematics - Straight Lines: If in a parallelogram $A B D C$ , the coordinates of $A$ , $B$ and $C$ are respectively $(1, 2)$ , $(3, 4)$ and $(2, 5)$ then the equation of the diagonal $A D$ is:

Select Answer:

Visualized Solution

Visualizing the Given Points

Forming the Parallelogram

Property of Diagonals

Midpoint of Diagonal $B C$

Calculating Midpoint $M$

Midpoint of Diagonal $A D$

Solving for $x$ -coordinate of $D$

Solving for $y$ -coordinate of $D$

Equation of a Line (Two-Point Form)

Substituting Coordinates

Calculating the Slope

Cross-Multiplication

Final Equation of Diagonal $A D$

The Sigma Insight: Various Forms of Equations of a Line

Analyzing the Setup

The Geometric Insight

The Calculation

The Final Line

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