Find the equation of the perpendicular bisector of the line segment joining the points `(1,0)` and `(3,5)`.

Answer

To find the equation of the perpendicular bisector of the line segment joining the points \( (1,0) \) and \( (3,5) \), we can follow these steps:

### Step 1: Find the Midpoint of the Line Segment
The midpoint \( M \) of the line segment joining the points \( A(1, 0) \) and \( B(3, 5) \) can be calculated using the midpoint formula:

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

Substituting the coordinates of points \( A \) and \( B \):

\[ M = \left( \frac{1 + 3}{2}, \frac{0 + 5}{2} \right) = \left( \frac{4}{2}, \frac{5}{2} \right) = (2, \frac{5}{2}) \]

### Step 2: Find the Slope of the Line Segment
Next, we need to find the slope \( m_1 \) of the line segment \( AB \) using the slope formula:

\[ m_1 = \frac{y_2 - y_1}{x_2 - x_1} \]

Substituting the coordinates of points \( A \) and \( B \):

\[ m_1 = \frac{5 - 0}{3 - 1} = \frac{5}{2} \]

### Step 3: Find the Slope of the Perpendicular Bisector
The slope \( m_2 \) of the perpendicular bisector is the negative reciprocal of \( m_1 \):

\[ m_2 = -\frac{1}{m_1} = -\frac{1}{\frac{5}{2}} = -\frac{2}{5} \]

### Step 4: Use the Point-Slope Form to Find the Equation
Now, we can use the point-slope form of the equation of a line, which is given by:

\[ y - y_1 = m(x - x_1) \]

Using the midpoint \( M(2, \frac{5}{2}) \) and the slope \( m_2 = -\frac{2}{5} \):

\[ y - \frac{5}{2} = -\frac{2}{5}(x - 2) \]

### Step 5: Simplify the Equation
Now, we will simplify this equation:

\[ y - \frac{5}{2} = -\frac{2}{5}x + \frac{4}{5} \]

Adding \( \frac{5}{2} \) to both sides:

\[ y = -\frac{2}{5}x + \frac{4}{5} + \frac{5}{2} \]

To combine \( \frac{4}{5} \) and \( \frac{5}{2} \), we need a common denominator, which is 10:

\[ \frac{5}{2} = \frac{25}{10}, \quad \frac{4}{5} = \frac{8}{10} \]

So,

\[ y = -\frac{2}{5}x + \frac{8}{10} + \frac{25}{10} = -\frac{2}{5}x + \frac{33}{10} \]

### Step 6: Rearranging to Standard Form
To convert this into standard form \( Ax + By = C \):

Multiply through by 10 to eliminate the fraction:

\[ 10y = -4x + 33 \]

Rearranging gives:

\[ 4x + 10y = 33 \]

### Final Answer
Thus, the equation of the perpendicular bisector of the line segment joining the points \( (1,0) \) and \( (3,5) \) is:

\[ 4x + 10y = 33 \]