`A(1,3)and C(7,5)` are two opposite vertices of a square. The equation of a side through A is

Options

A. `x+2y-7=0`
B. `x-2y+5=0`
C. `2x+y-5=0`
D. `2x-y+1=0`

Answer

To find the equation of a side of the square through the point A(1, 3), given that A(1, 3) and C(7, 5) are opposite vertices of the square, we will follow these steps:

### Step 1: Find the slope of the diagonal AC.
The slope \( m_{AC} \) of the line segment AC can be calculated using the formula:
\[ m_{AC} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - 3}{7 - 1} = \frac{2}{6} = \frac{1}{3} \]

**Hint:** Remember that the slope of a line is the change in y divided by the change in x.

### Step 2: Determine the slope of the sides of the square.
Since the sides of the square are perpendicular to the diagonal, the slope of the sides \( m \) can be found using the negative reciprocal of the slope of the diagonal:
\[ m = -\frac{1}{m_{AC}} = -\frac{1}{\frac{1}{3}} = -3 \]

**Hint:** The slopes of perpendicular lines multiply to -1.

### Step 3: Use point-slope form to write the equation of the line through A(1, 3).
Using the point-slope form of the equation of a line, which is:
\[ y - y_1 = m(x - x_1) \]
Substituting \( m = -3 \), \( x_1 = 1 \), and \( y_1 = 3 \):
\[ y - 3 = -3(x - 1) \]

**Hint:** The point-slope form is useful when you have a point and a slope.

### Step 4: Simplify the equation.
Expanding and simplifying the equation:
\[ y - 3 = -3x + 3 \]
\[ y = -3x + 6 \]

**Hint:** Always combine like terms to simplify your equation.

### Step 5: Rearranging to standard form.
To express the equation in standard form \( Ax + By = C \):
\[ 3x + y = 6 \]

**Hint:** Standard form is often preferred for equations of lines.

### Final Answer:
The equation of the side through A(1, 3) is:
\[ 3x + y = 6 \]

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