To solve this problem, let's follow these steps:
1. Find the slope of line [tex]\( \overrightarrow{A B} \)[/tex]:
- We are given the coordinates of points [tex]\( A \)[/tex] and [tex]\( B \)[/tex] as [tex]\( A(-10, -3) \)[/tex] and [tex]\( B(7, 14) \)[/tex].
- The formula for the slope ([tex]\( m \)[/tex]) of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Plugging in the coordinates of [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[
m = \frac{14 - (-3)}{7 - (-10)} = \frac{14 + 3}{7 + 10} = \frac{17}{17} = 1
\][/tex]
- Therefore, the slope of line [tex]\( \overrightarrow{A B} \)[/tex] is 1.
2. Find the slope of line [tex]\( \overleftrightarrow{C D} \)[/tex]:
- Since [tex]\( \overleftrightarrow{C D} \)[/tex] is perpendicular to [tex]\( \overrightarrow{A B} \)[/tex], the slope of [tex]\( \overleftrightarrow{C D} \)[/tex] will be the negative reciprocal of the slope of [tex]\( \overrightarrow{A B} \)[/tex]:
[tex]\[
\text{slope of } \overleftrightarrow{C D} = -\frac{1}{1} = -1
\][/tex]
3. Find the equation of line [tex]\( \overleftrightarrow{C D} \)[/tex]:
- The line passes through point [tex]\( C(5, 12) \)[/tex] and has a slope of [tex]\(-1\)[/tex].
- The equation of a line in slope-intercept form is [tex]\( y = mx + b \)[/tex].
- Plugging in the slope ([tex]\( m = -1 \)[/tex]) and the coordinates of [tex]\( C \)[/tex]:
[tex]\[
12 = -1 \cdot 5 + b
\][/tex]
[tex]\[
12 = -5 + b
\][/tex]
[tex]\[
b = 12 + 5 = 17
\][/tex]
- Therefore, the equation of the line [tex]\( \overleftrightarrow{C D} \)[/tex] is:
[tex]\[
y = -x + 17
\][/tex]
4. Find the [tex]\( x \)[/tex]-intercept of line [tex]\( \overleftrightarrow{C D} \)[/tex]:
- The [tex]\( x \)[/tex]-intercept is the point where [tex]\( y = 0 \)[/tex].
- Setting [tex]\( y = 0 \)[/tex] in the equation [tex]\( y = -x + 17 \)[/tex]:
[tex]\[
0 = -x + 17
\][/tex]
[tex]\[
x = 17
\][/tex]
- Therefore, the [tex]\( x \)[/tex]-intercept is [tex]\( (17, 0) \)[/tex].
5. Determine which point lies on [tex]\( \overleftrightarrow{C D} \)[/tex]:
- Given the [tex]\( x \)[/tex]-intercept determination, and knowing the structure of the line [tex]\( y = -x + 17 \)[/tex]:
- We can verify each option by substituting the [tex]\( x \)[/tex] value into the equation and checking if [tex]\( y \)[/tex] equals the given [tex]\( y \)[/tex] in each option.
- The point that lies on [tex]\( \overleftrightarrow{C D} \)[/tex] is [tex]\( (17, 0) \)[/tex].
Therefore, the correct answers are:
- The [tex]\( x \)[/tex]-intercept is [tex]\( (17, 0) \)[/tex].
- The point [tex]\( (17, 0) \)[/tex] lies on [tex]\( \overleftrightarrow{C D} \)[/tex].
