Find the information you're looking for at Westonci.ca, the trusted Q&A platform with a community of knowledgeable experts. Get quick and reliable solutions to your questions from a community of experienced experts on our platform. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
Let's solve the problem step by step.
1. Finding the slope of the line [tex]\(\overleftrightarrow{A B}\)[/tex]:
- The coordinates of points [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are [tex]\((14, -1)\)[/tex] and [tex]\((2, 1)\)[/tex], respectively.
- The formula for the slope [tex]\( m_{AB} \)[/tex] of the line passing through these points is given by:
[tex]\[ m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - (-1)}{2 - 14} = \frac{1 + 1}{2 - 14} = \frac{2}{-12} = -\frac{1}{6} \][/tex]
2. Finding the y-intercept of the line [tex]\(\overleftrightarrow{A B}\)[/tex]:
- The equation of the line in slope-intercept form is [tex]\( y = m_{AB} x + b_{AB} \)[/tex].
- Using the coordinates of point [tex]\(B\)[/tex], which is [tex]\((2, 1)\)[/tex]:
[tex]\[ 1 = -\frac{1}{6} \cdot 2 + b_{AB} \][/tex]
[tex]\[ 1 = -\frac{2}{6} + b_{AB} \][/tex]
[tex]\[ 1 = -\frac{1}{3} + b_{AB} \][/tex]
[tex]\[ b_{AB} = 1 + \frac{1}{3} = \frac{3}{3} + \frac{1}{3} = \frac{4}{3} \][/tex]
- Thus, the y-intercept of [tex]\(\overleftrightarrow{A B}\)[/tex] is [tex]\(\frac{4}{3}\)[/tex] or [tex]\(1.333333333\)[/tex].
3. Finding the equation of the line [tex]\(\overleftrightarrow{B C}\)[/tex]:
- Line [tex]\(\overleftrightarrow{B C}\)[/tex] is perpendicular to [tex]\(\overleftrightarrow{A B}\)[/tex], hence its 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]
- Using the point-slope form of the equation of the line and the coordinates of point [tex]\(B\)[/tex] (2, 1):
[tex]\[ y - 1 = 6(x - 2) \][/tex]
[tex]\[ y - 1 = 6x - 12 \][/tex]
[tex]\[ y = 6x - 12 + 1 \][/tex]
[tex]\[ y = 6x - 11 \][/tex]
- Therefore, the equation of the line [tex]\(\overleftrightarrow{B C}\)[/tex] is [tex]\( y = 6x - 11 \)[/tex].
4. Finding the x-coordinate of point [tex]\(C\)[/tex] when the y-coordinate of [tex]\(C\)[/tex] is 13:
[tex]\[ 13 = 6x - 11 \][/tex]
[tex]\[ 13 + 11 = 6x \][/tex]
[tex]\[ 24 = 6x \][/tex]
[tex]\[ x = \frac{24}{6} = 4 \][/tex]
- Thus, the x-coordinate of point [tex]\(C\)[/tex] is 4.
To summarize:
- The y-intercept of [tex]\(\overleftrightarrow{A B}\)[/tex] is [tex]\(1.333333333\)[/tex].
- The equation of [tex]\(\overleftrightarrow{B C}\)[/tex] is [tex]\( y = 6x - 11 \)[/tex].
- If the y-coordinate of point [tex]\(C\)[/tex] is 13, its x-coordinate is [tex]\(4\)[/tex].
So the correct answers for the boxes are:
1. [tex]\(\boxed{1.333333333}\)[/tex]
2. [tex]\(\boxed{6}\)[/tex]
3. [tex]\(\boxed{-11}\)[/tex]
4. [tex]\(\boxed{4}\)[/tex]
1. Finding the slope of the line [tex]\(\overleftrightarrow{A B}\)[/tex]:
- The coordinates of points [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are [tex]\((14, -1)\)[/tex] and [tex]\((2, 1)\)[/tex], respectively.
- The formula for the slope [tex]\( m_{AB} \)[/tex] of the line passing through these points is given by:
[tex]\[ m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - (-1)}{2 - 14} = \frac{1 + 1}{2 - 14} = \frac{2}{-12} = -\frac{1}{6} \][/tex]
2. Finding the y-intercept of the line [tex]\(\overleftrightarrow{A B}\)[/tex]:
- The equation of the line in slope-intercept form is [tex]\( y = m_{AB} x + b_{AB} \)[/tex].
- Using the coordinates of point [tex]\(B\)[/tex], which is [tex]\((2, 1)\)[/tex]:
[tex]\[ 1 = -\frac{1}{6} \cdot 2 + b_{AB} \][/tex]
[tex]\[ 1 = -\frac{2}{6} + b_{AB} \][/tex]
[tex]\[ 1 = -\frac{1}{3} + b_{AB} \][/tex]
[tex]\[ b_{AB} = 1 + \frac{1}{3} = \frac{3}{3} + \frac{1}{3} = \frac{4}{3} \][/tex]
- Thus, the y-intercept of [tex]\(\overleftrightarrow{A B}\)[/tex] is [tex]\(\frac{4}{3}\)[/tex] or [tex]\(1.333333333\)[/tex].
3. Finding the equation of the line [tex]\(\overleftrightarrow{B C}\)[/tex]:
- Line [tex]\(\overleftrightarrow{B C}\)[/tex] is perpendicular to [tex]\(\overleftrightarrow{A B}\)[/tex], hence its 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]
- Using the point-slope form of the equation of the line and the coordinates of point [tex]\(B\)[/tex] (2, 1):
[tex]\[ y - 1 = 6(x - 2) \][/tex]
[tex]\[ y - 1 = 6x - 12 \][/tex]
[tex]\[ y = 6x - 12 + 1 \][/tex]
[tex]\[ y = 6x - 11 \][/tex]
- Therefore, the equation of the line [tex]\(\overleftrightarrow{B C}\)[/tex] is [tex]\( y = 6x - 11 \)[/tex].
4. Finding the x-coordinate of point [tex]\(C\)[/tex] when the y-coordinate of [tex]\(C\)[/tex] is 13:
[tex]\[ 13 = 6x - 11 \][/tex]
[tex]\[ 13 + 11 = 6x \][/tex]
[tex]\[ 24 = 6x \][/tex]
[tex]\[ x = \frac{24}{6} = 4 \][/tex]
- Thus, the x-coordinate of point [tex]\(C\)[/tex] is 4.
To summarize:
- The y-intercept of [tex]\(\overleftrightarrow{A B}\)[/tex] is [tex]\(1.333333333\)[/tex].
- The equation of [tex]\(\overleftrightarrow{B C}\)[/tex] is [tex]\( y = 6x - 11 \)[/tex].
- If the y-coordinate of point [tex]\(C\)[/tex] is 13, its x-coordinate is [tex]\(4\)[/tex].
So the correct answers for the boxes are:
1. [tex]\(\boxed{1.333333333}\)[/tex]
2. [tex]\(\boxed{6}\)[/tex]
3. [tex]\(\boxed{-11}\)[/tex]
4. [tex]\(\boxed{4}\)[/tex]
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.
