To find the [tex]$x$[/tex]-intercept of the line [tex]$\overleftrightarrow{CD}$[/tex] and the coordinates of another point [tex]$D$[/tex] that lies on this line, we'll follow these steps:
1. Calculate the slope of line [tex]\(\overleftrightarrow{AB}\)[/tex]:
- The coordinates of points [tex]$A$[/tex] and [tex]$B$[/tex] are [tex]$(-10, -3)$[/tex] and [tex]$(7, 14)$[/tex], respectively.
- Slope of [tex]$\overleftrightarrow{AB}$[/tex], denoted as [tex]$m_{AB}$[/tex], can be calculated using the formula:
[tex]\[
m_{AB} = \frac{y_B - y_A}{x_B - x_A} = \frac{14 - (-3)}{7 - (-10)} = \frac{17}{17} = 1
\][/tex]
2. Determine the slope of line [tex]\(\overleftrightarrow{CD}\)[/tex] (perpendicular to [tex]\(\overleftrightarrow{AB}\)[/tex]):
- The slope of a line perpendicular to another is the negative reciprocal of the original slope. So, the slope of [tex]\(\overleftrightarrow{CD}\)[/tex], denoted as [tex]$m_{CD}$[/tex], is:
[tex]\[
m_{CD} = -\frac{1}{m_{AB}} = -1
\][/tex]
3. Formulate the equation of line [tex]\(\overleftrightarrow{CD}\)[/tex]:
- Line [tex]$\overleftrightarrow{CD}$[/tex] passes through point [tex]$C(5, 12)$[/tex] and has a slope of [tex]$-1$[/tex]. The equation of the line in slope-intercept form [tex]\(y = mx + b\)[/tex] can be determined as follows:
[tex]\[
y = -1 x + b
\][/tex]
- Substitute point [tex]$C(5, 12)$[/tex] into the equation to solve for [tex]$b$[/tex] (the intercept):
[tex]\[
12 = -1 \cdot 5 + b \implies 12 = -5 + b \implies b = 17
\][/tex]
- Thus, the equation of line [tex]\(\overleftrightarrow{CD}\)[/tex] is:
[tex]\[
y = -x + 17
\][/tex]
4. Find the [tex]$x$[/tex]-intercept of line [tex]\(\overleftrightarrow{CD}\)[/tex]:
- The [tex]$x$[/tex]-intercept is where the line crosses the [tex]$x$[/tex]-axis (i.e., where [tex]$y = 0$[/tex]).
[tex]\[
0 = -x + 17 \implies x = 17
\][/tex]
- So, the [tex]$x$[/tex]-intercept of [tex]$\overleftrightarrow{CD}$[/tex] is 17.
5. Determine a point on [tex]\(\overleftrightarrow{CD}\)[/tex] by substituting a specific [tex]$x$[/tex]\-value:
- For simplicity, choose [tex]\(x = 0\)[/tex]:
- Substitute [tex]$x = 0$[/tex] into the equation [tex]\(y = -x + 17\)[/tex]:
[tex]\[
y = -0 + 17 = 17
\][/tex]
- Thus, the point [tex]\(D\)[/tex] when [tex]\(x = 0\)[/tex] is [tex]\((0, 17)\)[/tex].
Based on this solution, the [tex]$x$[/tex]-intercept of [tex]\(\overleftrightarrow{CD}\)[/tex] is [tex]\(\boxed{17}\)[/tex] and a point that lies on [tex]\(\overleftrightarrow{CD}\)[/tex] other than [tex]$C$[/tex] is [tex]\(\boxed{(0, 17)}\)[/tex].
