To solve this problem, we need to find several values based on the coordinates of points [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
1. Determine the slope of line [tex]\(\overleftrightarrow{A B}\)[/tex]:
The coordinates of [tex]\( A \)[/tex] are [tex]\((14, -1)\)[/tex] and the coordinates of [tex]\( B \)[/tex] are [tex]\((2, 1)\)[/tex].
The formula for the slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Applying the coordinates of [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[
\text{slope}_{AB} = \frac{1 - (-1)}{2 - 14} = \frac{2}{-12} = -\frac{1}{6}
\][/tex]
So, the slope of [tex]\(\overleftrightarrow{A B}\)[/tex] is [tex]\(-\frac{1}{6}\)[/tex].
2. Find the y-intercept of [tex]\(\overleftrightarrow{A B}\)[/tex]:
The equation of a line in slope-intercept form is [tex]\( y = mx + b \)[/tex].
Using point [tex]\( B \)[/tex] ([tex]\(2, 1\)[/tex]) to find [tex]\( b \)[/tex]:
[tex]\[
1 = -\frac{1}{6} \cdot 2 + b
\][/tex]
[tex]\[
1 = -\frac{1}{3} + b
\][/tex]
[tex]\[
b = 1 + \frac{1}{3} = \frac{4}{3}
\][/tex]
So, the y-intercept of [tex]\(\overleftrightarrow{A B}\)[/tex] is [tex]\( \frac{4}{3} \)[/tex] or approximately [tex]\( 1.3333333333333333 \)[/tex].
3. Determine the slope of line [tex]\(\overleftrightarrow{B C}\)[/tex]:
Since [tex]\(\overleftrightarrow{B C}\)[/tex] is perpendicular to [tex]\(\overleftrightarrow{A B}\)[/tex], the slope of [tex]\(\overleftrightarrow{B C}\)[/tex] is the negative reciprocal of the slope of [tex]\(\overleftrightarrow{A B}\)[/tex].
The slope of [tex]\(\overleftrightarrow{A B}\)[/tex] is [tex]\(-\frac{1}{6}\)[/tex], so:
[tex]\[
\text{slope}_{BC} = -1 \div -\frac{1}{6} = 6
\][/tex]
Therefore, the slope of [tex]\(\overleftrightarrow{B C}\)[/tex] is 6.
4. Find the y-intercept of [tex]\(\overleftrightarrow{B C}\)[/tex]:
Using point [tex]\( B \)[/tex] ([tex]\(2, 1\)[/tex]) to find [tex]\( b \)[/tex]:
[tex]\[
1 = 6 \cdot 2 + b
\][/tex]
[tex]\[
1 = 12 + b
\][/tex]
[tex]\[
b = 1 - 12 = -11
\][/tex]
So, the y-intercept of [tex]\(\overleftrightarrow{B C}\)[/tex] is [tex]\( -11 \)[/tex].
5. Find the x-coordinate of point [tex]\( C \)[/tex] if its y-coordinate is 13:
Using the equation of [tex]\(\overleftrightarrow{B C}\)[/tex]:
[tex]\[
y = 6x - 11
\][/tex]
Substitute [tex]\( y = 13 \)[/tex]:
[tex]\[
13 = 6x - 11
\][/tex]
[tex]\[
24 = 6x
\][/tex]
[tex]\[
x = \frac{24}{6} = 4
\][/tex]
So, the solutions are:
- The y-intercept of [tex]\(\overleftrightarrow{A B}\)[/tex] is [tex]\( \frac{4}{3} \)[/tex] or [tex]\( 1.3333333333333333 \)[/tex].
- The equation of [tex]\(\overleftrightarrow{B C}\)[/tex] is [tex]\( y = 6x - 11 \)[/tex].
- The x-coordinate of point [tex]\( C \)[/tex] is [tex]\( 4 \)[/tex].
