To find the axes intercepts of the perpendicular bisector of the line segment [tex]\([AB]\)[/tex] where [tex]\(A\)[/tex] is [tex]\((1, 2)\)[/tex] and [tex]\(B\)[/tex] is [tex]\((9, 18)\)[/tex], let's break down the process step-by-step:
### Step 1: Calculate the Midpoint of [tex]\( [AB] \)[/tex]
The midpoint of a line segment with endpoints [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] can be found using the midpoint formula:
[tex]\[
\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
For our points [tex]\( A(1, 2) \)[/tex] and [tex]\( B(9, 18) \)[/tex]:
[tex]\[
\text{Midpoint} = \left( \frac{1 + 9}{2}, \frac{2 + 18}{2} \right) = (5.0, 10.0)
\][/tex]
### Step 2: Calculate the Slope of [tex]\( [AB] \)[/tex]
The slope of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
\text{Slope}_{AB} = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
For [tex]\( A(1, 2) \)[/tex] and [tex]\( B(9, 18) \)[/tex]:
[tex]\[
\text{Slope}_{AB} = \frac{18 - 2}{9 - 1} = \frac{16}{8} = 2.0
\][/tex]
### Step 3: Calculate the Slope of the Perpendicular Bisector
The slope of the perpendicular bisector is the negative reciprocal of the slope of [tex]\( [AB] \)[/tex]:
[tex]\[
\text{Slope}_{\text{perpendicular}} = -\frac{1}{\text{Slope}_{AB}} = -\frac{1}{2.0} = -0.5
\][/tex]
### Step 4: Find the Equation of the Perpendicular Bisector
To find the equation of the perpendicular bisector, we use the point-slope form of a line equation:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
where [tex]\((x_1, y_1)\)[/tex] is the midpoint [tex]\((5.0, 10.0)\)[/tex] and [tex]\(m\)[/tex] is the slope of the perpendicular bisector [tex]\(-0.5\)[/tex]:
[tex]\[
y - 10.0 = -0.5(x - 5.0)
\][/tex]
Solving for [tex]\(y\)[/tex] to get the slope-intercept form:
[tex]\[
y = -0.5x + 12.5
\][/tex]
This is the equation of the perpendicular bisector:
[tex]\[
y = -0.5x + 12.5
\][/tex]
### Step 5: Find the x-intercept
To find the x-intercept, set [tex]\( y = 0 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[
0 = -0.5x + 12.5 \\
0.5x = 12.5 \\
x = \frac{12.5}{0.5} \\
x = 25.0
\][/tex]
Thus, the x-intercept is [tex]\( (25.0, 0) \)[/tex].
### Step 6: Find the y-intercept
To find the y-intercept, set [tex]\( x = 0 \)[/tex] and solve for [tex]\( y \)[/tex]:
[tex]\[
y = -0.5(0) + 12.5 \\
y = 12.5
\][/tex]
Thus, the y-intercept is [tex]\( (0, 12.5) \)[/tex].
### Conclusion
The coordinates of the axis intercepts of the perpendicular bisector of [tex]\([AB]\)[/tex] are:
- [tex]\( x \)[/tex]-intercept: [tex]\( (25.0, 0) \)[/tex]
- [tex]\( y \)[/tex]-intercept: [tex]\( (0, 12.5) \)[/tex]
