Question

Line AB
𝐴
𝐵
contains point A(−4, 1)
𝐴
(
−
4
,

1
)
and point B (−1, 3)
𝐵

(
−
1
,

3
)

Answer 1

To find information about the line segment \(AB\) that passes through the points \(A(-4, 1)\) and \(B(-1, 3)\), we can calculate the slope of the line, the equation of the line, and the length of the line segment.

### 1. Calculate the Slope
The slope \(m\) of the line connecting points \(A(x_1, y_1)\) and \(B(x_2, y_2)\) can be calculated with the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Substituting the coordinates of points \(A\) and \(B\):
\[
m = \frac{3 - 1}{-1 - (-4)} = \frac{2}{3}
\]

### 2. Write the Equation of the Line
Using the point-slope form of a line equation \(y - y_1 = m(x - x_1)\):
\[
y - 1 = \frac{2}{3}(x + 4)
\]

Distributing the slope:
\[
y - 1 = \frac{2}{3}x + \frac{8}{3}
\]

Now, add 1 (or \(\frac{3}{3}\)) to both sides to isolate \(y\):
\[
y = \frac{2}{3}x + \frac{8}{3} + \frac{3}{3}
\]
\[
y = \frac{2}{3}x + \frac{11}{3}
\]

### 3. Calculate the Length of the Segment
The length \(d\) of the line segment can be found using the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Substituting the coordinates of points \(A\) and \(B\):
\[
d = \sqrt{(-1 - (-4))^2 + (3 - 1)^2}
\]
\[
d = \sqrt{(-1 + 4)^2 + (2)^2} = \sqrt{(3)^2 + (2)^2} = \sqrt{9 + 4} = \sqrt{13}
\]

### Summary
- The slope of line segment \(AB\) is \(\frac{2}{3}\).
- The equation of line segment \(AB\) is \(y = \frac{2}{3}x + \frac{11}{3}\).
- The length of line segment \(AB\) is \(\sqrt{13}\).