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