Westonci.ca is the best place to get answers to your questions, provided by a community of experienced and knowledgeable experts. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
Let's solve the problem step-by-step:
1. Determine the slope of line \( \overleftrightarrow{A B} \):
The coordinates are \( A = (14, -1) \) and \( B = (2, 1) \).
The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Here,
[tex]\[ m = \frac{1 - (-1)}{2 - 14} = \frac{2}{-12} = -\frac{1}{6} \][/tex]
So, the slope of \( \overleftrightarrow{A B} \) is \(-\frac{1}{6}\).
2. Find the y-intercept of line \( \overleftrightarrow{A B} \):
The equation of a line in slope-intercept form is \( y = mx + c \). We already have \( m = -\frac{1}{6} \).
To find \( c \) (the y-intercept), we use the point \( B \) which is \((2, 1)\):
[tex]\[ 1 = -\frac{1}{6} \cdot 2 + c \][/tex]
Simplifying, we get:
[tex]\[ 1 = -\frac{1}{3} + c \implies c = 1 + \frac{1}{3} = \frac{4}{3} \][/tex]
So, the y-intercept is \(\frac{4}{3}\) and the equation of \( \overleftrightarrow{A B} \) is:
[tex]\[ y = -\frac{1}{6}x + \frac{4}{3} \][/tex]
3. Find the slope of line \( \overleftrightarrow{B C} \):
Since \( \overleftrightarrow{B C} \) is perpendicular to \( \overleftrightarrow{A B} \), the product of their slopes is \(-1\).
Let \( m_{BC} \) be the slope of \( \overleftrightarrow{B C} \). Then:
[tex]\[ m_{BC} \cdot \left( -\frac{1}{6} \right) = -1 \implies m_{BC} = 6 \][/tex]
Thus, the slope of line \( \overleftrightarrow{B C} \) is \( 6 \).
4. Determine the x-coordinate of point \( C \) given its y-coordinate is 13:
The equation of \( \overleftrightarrow{B C} \) can be written using the slope-intercept form and the point \( B = (2, 1) \):
[tex]\[ y - 1 = 6(x - 2) \][/tex]
Given \( y = 13 \):
[tex]\[ 13 - 1 = 6(x - 2) \][/tex]
Simplifying this equation:
[tex]\[ 12 = 6(x - 2) \implies 2 = x - 2 \implies x = 4 \][/tex]
So, the x-coordinate of point \( C \) is \( 4 \).
Now, let's fill in the boxes in the original problem statement accordingly:
If the coordinates of \( A \) and \( B \) are \( (14, -1) \) and \( (2, 1) \), respectively, the \( y \)-intercept of \( \overleftrightarrow{A B} \) is \( y = -\frac{1}{6}x + \frac{4}{3} \).
If the [tex]\( y \)[/tex]-coordinate of point [tex]\( C \)[/tex] is [tex]\( 13 \)[/tex], its [tex]\( x \)[/tex]-coordinate is [tex]\( 4 \)[/tex].
1. Determine the slope of line \( \overleftrightarrow{A B} \):
The coordinates are \( A = (14, -1) \) and \( B = (2, 1) \).
The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Here,
[tex]\[ m = \frac{1 - (-1)}{2 - 14} = \frac{2}{-12} = -\frac{1}{6} \][/tex]
So, the slope of \( \overleftrightarrow{A B} \) is \(-\frac{1}{6}\).
2. Find the y-intercept of line \( \overleftrightarrow{A B} \):
The equation of a line in slope-intercept form is \( y = mx + c \). We already have \( m = -\frac{1}{6} \).
To find \( c \) (the y-intercept), we use the point \( B \) which is \((2, 1)\):
[tex]\[ 1 = -\frac{1}{6} \cdot 2 + c \][/tex]
Simplifying, we get:
[tex]\[ 1 = -\frac{1}{3} + c \implies c = 1 + \frac{1}{3} = \frac{4}{3} \][/tex]
So, the y-intercept is \(\frac{4}{3}\) and the equation of \( \overleftrightarrow{A B} \) is:
[tex]\[ y = -\frac{1}{6}x + \frac{4}{3} \][/tex]
3. Find the slope of line \( \overleftrightarrow{B C} \):
Since \( \overleftrightarrow{B C} \) is perpendicular to \( \overleftrightarrow{A B} \), the product of their slopes is \(-1\).
Let \( m_{BC} \) be the slope of \( \overleftrightarrow{B C} \). Then:
[tex]\[ m_{BC} \cdot \left( -\frac{1}{6} \right) = -1 \implies m_{BC} = 6 \][/tex]
Thus, the slope of line \( \overleftrightarrow{B C} \) is \( 6 \).
4. Determine the x-coordinate of point \( C \) given its y-coordinate is 13:
The equation of \( \overleftrightarrow{B C} \) can be written using the slope-intercept form and the point \( B = (2, 1) \):
[tex]\[ y - 1 = 6(x - 2) \][/tex]
Given \( y = 13 \):
[tex]\[ 13 - 1 = 6(x - 2) \][/tex]
Simplifying this equation:
[tex]\[ 12 = 6(x - 2) \implies 2 = x - 2 \implies x = 4 \][/tex]
So, the x-coordinate of point \( C \) is \( 4 \).
Now, let's fill in the boxes in the original problem statement accordingly:
If the coordinates of \( A \) and \( B \) are \( (14, -1) \) and \( (2, 1) \), respectively, the \( y \)-intercept of \( \overleftrightarrow{A B} \) is \( y = -\frac{1}{6}x + \frac{4}{3} \).
If the [tex]\( y \)[/tex]-coordinate of point [tex]\( C \)[/tex] is [tex]\( 13 \)[/tex], its [tex]\( x \)[/tex]-coordinate is [tex]\( 4 \)[/tex].
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.
