A line perpendicular to the line segment joining the points A(1,0)*A*(1,0) and B(2,3)*B*(2,3), divides it at C*C* in the ratio of 1:31:3. Then the equation of the line is*?*

Let's solve this step by step.

We are given:
- Points \( A(1, 0) \) and \( B(2, 3) \).
- The line that is perpendicular to the line segment joining \( A \) and \( B \) divides the segment at \( C \) in the ratio \( 1:3 \).

### Step 1: Find the coordinates of point \( C \)
The point \( C \) divides the line segment joining \( A \) and \( B \) in the ratio \( 1:3 \). To find the coordinates of \( C \), we use the section formula:
\[
C(x_C, y_C) = \left( \frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n} \right)
\]
where \( A(x_1, y_1) = (1, 0) \), \( B(x_2, y_2) = (2, 3) \), and the ratio is \( m:n = 1:3 \).

So, 
\[
x_C = \frac{1 \cdot 2 + 3 \cdot 1}{1+3} = \frac{2 + 3}{4} = \frac{5}{4}
\]
\[
y_C = \frac{1 \cdot 3 + 3 \cdot 0}{1+3} = \frac{3 + 0}{4} = \frac{3}{4}
\]

Thus, the coordinates of \( C \) are \( C\left( \frac{5}{4}, \frac{3}{4} \right) \).

### Step 2: Find the slope of line \( AB \)
The slope of the line segment joining \( A \) and \( B \) is given by:
\[
\text{slope of } AB = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 0}{2 - 1} = 3
\]

### Step 3: Find the slope of the perpendicular line
The line we are looking for is perpendicular to \( AB \), so the slope of this perpendicular line will be the negative reciprocal of the slope of \( AB \).

The slope of the perpendicular line is:
\[
\text{slope of perpendicular line} = -\frac{1}{3}
\]

### Step 4: Use point-slope form to find the equation of the perpendicular line
Now that we have the slope of the perpendicular line and a point on the line, \( C\left( \frac{5}{4}, \frac{3}{4} \right) \), we can use the point-slope form of the equation of a line:
\[
y - y_1 = m(x - x_1)
\]
Substitute \( m = -\frac{1}{3} \), \( x_1 = \frac{5}{4} \), and \( y_1 = \frac{3}{4} \):
\[
y - \frac{3}{4} = -\frac{1}{3}(x - \frac{5}{4})
\]

### Step 5: Simplify the equation
First, distribute the slope:
\[
y - \frac{3}{4} = -\frac{1}{3}x + \frac{5}{12}
\]
Now, add \( \frac{3}{4} \) to both sides to isolate \( y \):
\[
y = -\frac{1}{3}x + \frac{5}{12} + \frac{3}{4}
\]
Convert \( \frac{3}{4} \) to a fraction with denominator 12:
\[
\frac{3}{4} = \frac{9}{12}
\]
So:
\[
y = -\frac{1}{3}x + \frac{5}{12} + \frac{9}{12} = -\frac{1}{3}x + \frac{14}{12} = -\frac{1}{3}x + \frac{7}{6}
\]

### Final Answer:
The equation of the line is:
\[
y = -\frac{1}{3}x + \frac{7}{6}
\]

Would you like further details or clarifications on any step?

---

Here are 5 related questions for further exploration:

1. How would the equation change if the ratio of division were \( 2:1 \) instead of \( 1:3 \)?
2. What is the general formula for the slope of a line perpendicular to a given line?
3. Can you find the midpoint of the segment joining points \( A \) and \( B \)?
4. How would you find the distance between two points in 2D space?
5. What is the geometric significance of the section formula?

**Tip:** Always check if two lines are perpendicular by verifying that the product of their slopes equals -1.

A line perpendicular to the line segment joining the points A(1,0) and B(2,3) divides it at C in the ratio of 1:3. Then the equation of the line is?

This problem explores how to find the equation of a line that is perpendicular to the line segment joining points A(1,0) and B(2,3), dividing it at point C in the ratio 1:3. Using concepts from coordinate geometry, the section formula, and slope, this detailed solution is suitable for students in Grades 10-12.

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