Welcome to Westonci.ca, the place where your questions are answered by a community of knowledgeable contributors. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
Let's solve the problem step-by-step.
First, we are given the coordinates of points [tex]\( A \)[/tex] and [tex]\( B \)[/tex] as [tex]\( A(14, -1) \)[/tex] and [tex]\( B(2, 1) \)[/tex] respectively.
1. Calculate the slope of [tex]\(\overleftrightarrow{AB}\)[/tex]:
The formula for the slope [tex]\( m \)[/tex] between 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]
Substituting in the given coordinates:
[tex]\[ m_{AB} = \frac{1 - (-1)}{2 - 14} = \frac{2}{-12} = -\frac{1}{6} \][/tex]
2. Determine the y-intercept of [tex]\(\overleftrightarrow{AB}\)[/tex]:
The equation of the line can be written as [tex]\( y = mx + c \)[/tex], where [tex]\( c \)[/tex] is the y-intercept. Using the slope [tex]\( m_{AB} = -\frac{1}{6} \)[/tex] and point [tex]\( B(2, 1) \)[/tex]:
[tex]\[ 1 = \left( -\frac{1}{6} \right) \cdot 2 + c \][/tex]
[tex]\[ 1 = -\frac{2}{6} + c \][/tex]
[tex]\[ 1 = -\frac{1}{3} + c \][/tex]
Adding [tex]\(\frac{1}{3}\)[/tex] to both sides:
[tex]\[ c = 1 + \frac{1}{3} = \frac{3}{3} + \frac{1}{3} = \frac{4}{3} \][/tex]
So, the y-intercept [tex]\( c \)[/tex] of [tex]\(\overleftrightarrow{AB}\)[/tex] is:
[tex]\[ c = \frac{4}{3} \approx 1.333 \][/tex]
3. The equation of [tex]\(\overleftrightarrow{AB}\)[/tex]:
Now that we have the slope [tex]\( m_{AB} \)[/tex] and the y-intercept [tex]\( c \)[/tex]:
[tex]\[ y = -\frac{1}{6}x + \frac{4}{3} \][/tex]
4. Calculate the slope of [tex]\(\overleftrightarrow{BC}\)[/tex]:
Since [tex]\(\overleftrightarrow{AB}\)[/tex] and [tex]\(\overleftrightarrow{BC}\)[/tex] form a right angle, the slope [tex]\( m_{BC} \)[/tex] is the negative reciprocal of [tex]\( m_{AB} \)[/tex]:
[tex]\[ m_{BC} = -\frac{1}{m_{AB}} = -\frac{1}{-\frac{1}{6}} = 6 \][/tex]
5. Determine the x-coordinate of point [tex]\( C \)[/tex]:
Given the y-coordinate of point [tex]\( C \)[/tex] is [tex]\( 13 \)[/tex], use the point-slope form of the line [tex]\(\overleftrightarrow{BC}\)[/tex] passing through point [tex]\( B(2, 1)\)[/tex]:
[tex]\[ y - 1 = m_{BC}(x - 2) \][/tex]
Substituting [tex]\( y = 13 \)[/tex] and [tex]\( m_{BC} = 6 \)[/tex]:
[tex]\[ 13 - 1 = 6(x - 2) \][/tex]
[tex]\[ 12 = 6(x - 2) \][/tex]
[tex]\[ 12 = 6x - 12 \][/tex]
Adding 12 to both sides:
[tex]\[ 24 = 6x \][/tex]
Dividing both sides by 6:
[tex]\[ x = 4 \][/tex]
So, the answers are:
- The [tex]\( y \)[/tex]-intercept of [tex]\(\overleftrightarrow{AB}\)[/tex] is approximately [tex]\( 1.333 \)[/tex].
- The equation of [tex]\(\overleftrightarrow{AB}\)[/tex] is [tex]\( y = -\frac{1}{6}x + \frac{4}{3} \)[/tex].
- The [tex]\( x \)[/tex]-coordinate of point [tex]\( C \)[/tex] is [tex]\( 4 \)[/tex].
First, we are given the coordinates of points [tex]\( A \)[/tex] and [tex]\( B \)[/tex] as [tex]\( A(14, -1) \)[/tex] and [tex]\( B(2, 1) \)[/tex] respectively.
1. Calculate the slope of [tex]\(\overleftrightarrow{AB}\)[/tex]:
The formula for the slope [tex]\( m \)[/tex] between 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]
Substituting in the given coordinates:
[tex]\[ m_{AB} = \frac{1 - (-1)}{2 - 14} = \frac{2}{-12} = -\frac{1}{6} \][/tex]
2. Determine the y-intercept of [tex]\(\overleftrightarrow{AB}\)[/tex]:
The equation of the line can be written as [tex]\( y = mx + c \)[/tex], where [tex]\( c \)[/tex] is the y-intercept. Using the slope [tex]\( m_{AB} = -\frac{1}{6} \)[/tex] and point [tex]\( B(2, 1) \)[/tex]:
[tex]\[ 1 = \left( -\frac{1}{6} \right) \cdot 2 + c \][/tex]
[tex]\[ 1 = -\frac{2}{6} + c \][/tex]
[tex]\[ 1 = -\frac{1}{3} + c \][/tex]
Adding [tex]\(\frac{1}{3}\)[/tex] to both sides:
[tex]\[ c = 1 + \frac{1}{3} = \frac{3}{3} + \frac{1}{3} = \frac{4}{3} \][/tex]
So, the y-intercept [tex]\( c \)[/tex] of [tex]\(\overleftrightarrow{AB}\)[/tex] is:
[tex]\[ c = \frac{4}{3} \approx 1.333 \][/tex]
3. The equation of [tex]\(\overleftrightarrow{AB}\)[/tex]:
Now that we have the slope [tex]\( m_{AB} \)[/tex] and the y-intercept [tex]\( c \)[/tex]:
[tex]\[ y = -\frac{1}{6}x + \frac{4}{3} \][/tex]
4. Calculate the slope of [tex]\(\overleftrightarrow{BC}\)[/tex]:
Since [tex]\(\overleftrightarrow{AB}\)[/tex] and [tex]\(\overleftrightarrow{BC}\)[/tex] form a right angle, the slope [tex]\( m_{BC} \)[/tex] is the negative reciprocal of [tex]\( m_{AB} \)[/tex]:
[tex]\[ m_{BC} = -\frac{1}{m_{AB}} = -\frac{1}{-\frac{1}{6}} = 6 \][/tex]
5. Determine the x-coordinate of point [tex]\( C \)[/tex]:
Given the y-coordinate of point [tex]\( C \)[/tex] is [tex]\( 13 \)[/tex], use the point-slope form of the line [tex]\(\overleftrightarrow{BC}\)[/tex] passing through point [tex]\( B(2, 1)\)[/tex]:
[tex]\[ y - 1 = m_{BC}(x - 2) \][/tex]
Substituting [tex]\( y = 13 \)[/tex] and [tex]\( m_{BC} = 6 \)[/tex]:
[tex]\[ 13 - 1 = 6(x - 2) \][/tex]
[tex]\[ 12 = 6(x - 2) \][/tex]
[tex]\[ 12 = 6x - 12 \][/tex]
Adding 12 to both sides:
[tex]\[ 24 = 6x \][/tex]
Dividing both sides by 6:
[tex]\[ x = 4 \][/tex]
So, the answers are:
- The [tex]\( y \)[/tex]-intercept of [tex]\(\overleftrightarrow{AB}\)[/tex] is approximately [tex]\( 1.333 \)[/tex].
- The equation of [tex]\(\overleftrightarrow{AB}\)[/tex] is [tex]\( y = -\frac{1}{6}x + \frac{4}{3} \)[/tex].
- The [tex]\( x \)[/tex]-coordinate of point [tex]\( C \)[/tex] is [tex]\( 4 \)[/tex].
Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.
