Is the line through `(-2,3) and (4,1)` perpendicular to the line `3x =y + 1?` Does the line `3x = y +1` bisect the join of `(-2,3) and (4,1)?`

Answer

To solve the problem, we will follow these steps:

### Step 1: Find the slope of the line through points (-2, 3) and (4, 1).

The formula for the slope (m) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

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

Here, we have:
- \((x_1, y_1) = (-2, 3)\)
- \((x_2, y_2) = (4, 1)\)

Substituting the values into the formula:

\[ m_{AB} = \frac{1 - 3}{4 - (-2)} = \frac{-2}{4 + 2} = \frac{-2}{6} = -\frac{1}{3} \]

### Step 2: Find the slope of the line given by the equation \(3x = y + 1\).

First, we rearrange the equation into the slope-intercept form \(y = mx + b\):

\[ y = 3x - 1 \]

From this, we can see that the slope \(m_2\) of this line is:

\[ m_2 = 3 \]

### Step 3: Check if the two lines are perpendicular.

Two lines are perpendicular if the product of their slopes is \(-1\):

\[ m_{AB} \cdot m_2 = -\frac{1}{3} \cdot 3 = -1 \]

Since the product of the slopes is \(-1\), the lines are perpendicular.

### Step 4: Find the midpoint of the line segment joining points (-2, 3) and (4, 1).

The midpoint \(M\) of a line segment joining two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

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

Substituting the values:

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

### Step 5: Check if the line \(3x = y + 1\) bisects the line segment.

To check if the line bisects the segment, we need to see if the midpoint \(M(1, 2)\) satisfies the equation \(3x = y + 1\).

Substituting \(x = 1\) and \(y = 2\) into the equation:

\[ 3(1) = 2 + 1 \]

Calculating both sides:

\[ 3 = 3 \]

Since both sides are equal, the midpoint lies on the line.

### Conclusion:

1. The line through points (-2, 3) and (4, 1) is **perpendicular** to the line \(3x = y + 1\).
2. The line \(3x = y + 1\) **bisects** the line segment joining the points (-2, 3) and (4, 1).