Firstly, to find the x-intercept of the line [tex]\(\overleftrightarrow{C D}\)[/tex], which is perpendicular to [tex]\(\overleftrightarrow{A B}\)[/tex] and passes through point [tex]\(C(5,12)\)[/tex], we need to follow several steps:
1. Calculate the slope of [tex]\(\overleftrightarrow{A B}\)[/tex] using the coordinates of points [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:
- [tex]\(A = (-10, -3)\)[/tex]
- [tex]\(B = (7, 14)\)[/tex]
2. The slope formula is given by:
[tex]\[
\text{slope}_{AB} = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
3. Plugging in the coordinates of [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:
[tex]\[
\text{slope}_{AB} = \frac{14 - (-3)}{7 - (-10)} = \frac{14 + 3}{7 + 10} = \frac{17}{17} = 1.0
\][/tex]
4. Since [tex]\(\overleftrightarrow{C D}\)[/tex] is perpendicular to [tex]\(\overleftrightarrow{A B}\)[/tex], the slope of [tex]\(\overleftrightarrow{C D}\)[/tex] will be the negative reciprocal of the slope of [tex]\(\overleftrightarrow{A B}\)[/tex]:
[tex]\[
\text{slope}_{CD} = -\frac{1}{\text{slope}_{AB}} = -\frac{1}{1} = -1.0
\][/tex]
5. The equation of the line in point-slope form passing through [tex]\(C(5,12)\)[/tex] with slope [tex]\(-1.0\)[/tex] is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
6. Substituting [tex]\(C(5, 12)\)[/tex] and [tex]\(m = -1.0\)[/tex]:
[tex]\[
y - 12 = -1.0(x - 5)
\][/tex]
[tex]\[
y - 12 = -x + 5
\][/tex]
[tex]\[
y = -x + 17
\][/tex]
7. To find the x-intercept, set [tex]\(y = 0\)[/tex]:
[tex]\[
0 = -x + 17
\][/tex]
[tex]\[
x = 17
\][/tex]
Hence, the x-intercept of the line [tex]\( \overleftrightarrow{C D} \)[/tex] is [tex]\((17, 0)\)[/tex].
So, the correct answer from the drop-down menu is:
[tex]\[
(17,0)
\][/tex]
