The equation of one diagonal of a square is `2x+y=6` and its one vertex is `(4,3)`. Find the equation of other diagonal.

Answer

To find the equation of the other diagonal of the square given one diagonal's equation and a vertex, we can follow these steps:

### Step 1: Verify if the given vertex lies on the diagonal
We need to check if the point (4, 3) satisfies the equation of the diagonal \(2x + y = 6\).

Substituting \(x = 4\) and \(y = 3\) into the equation:
\[ 2(4) + 3 = 8 + 3 = 11 \neq 6 \]
So, the point (4, 3) does not lie on the diagonal \(2x + y = 6\).

### Step 2: Find the midpoint of the diagonal
Since the vertex (4, 3) does not lie on the diagonal, we need to find another vertex of the square that does lie on the diagonal. Let’s denote the other vertex of the square as \(A(x_1, y_1)\). The midpoint \(M\) of the diagonal can be found using the formula:
\[ M = \left( \frac{x_1 + 4}{2}, \frac{y_1 + 3}{2} \right) \]

### Step 3: Find the slope of the diagonal
The slope of the line \(2x + y = 6\) can be found by rewriting it in slope-intercept form \(y = mx + b\):
\[ y = -2x + 6 \]
Thus, the slope \(m_1\) of the diagonal is \(-2\).

### Step 4: Find the slope of the other diagonal
The diagonals of a square are perpendicular to each other. Therefore, the slope \(m_2\) of the other diagonal will be the negative reciprocal of \(m_1\):
\[ m_2 = \frac{1}{2} \]

### Step 5: Use the point-slope form to find the equation of the other diagonal
Using the point-slope form of the line equation \(y - y_1 = m(x - x_1)\) with the vertex (4, 3) and the slope \(\frac{1}{2}\):
\[ y - 3 = \frac{1}{2}(x - 4) \]

### Step 6: Simplify the equation
Expanding and simplifying the equation:
\[ y - 3 = \frac{1}{2}x - 2 \]
\[ y = \frac{1}{2}x + 1 \]

### Final Equation
Thus, the equation of the other diagonal is:
\[ \boxed{y = \frac{1}{2}x + 1} \]

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