Explore Westonci.ca, the premier Q&A site that helps you find precise answers to your questions, no matter the topic. Experience the convenience of getting accurate answers to your questions from a dedicated community of professionals. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
To find the equation of the line [tex]\(\overleftrightarrow{BC}\)[/tex] that forms a right angle with the line [tex]\(\overleftrightarrow{AB}\)[/tex] at point [tex]\(B\)[/tex], follow these steps:
1. Determine the slope of line [tex]\(\overleftrightarrow{AB}\)[/tex]:
The coordinates of points [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are given as [tex]\(A = (-3, -1)\)[/tex] and [tex]\(B = (4, 4)\)[/tex].
The formula to find 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 the coordinates of points [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:
[tex]\[ m_{AB} = \frac{4 - (-1)}{4 - (-3)} = \frac{4 + 1}{4 + 3} = \frac{5}{7} \approx 0.7142857142857143 \][/tex]
2. Calculate the slope of line [tex]\(\overleftrightarrow{BC}\)[/tex]:
Since [tex]\(\overleftrightarrow{BC}\)[/tex] is perpendicular to [tex]\(\overleftrightarrow{AB}\)[/tex] at point [tex]\(B\)[/tex], the slope of [tex]\(\overleftrightarrow{BC}\)[/tex] will be the negative reciprocal of the slope of [tex]\(\overleftrightarrow{AB}\)[/tex]. If [tex]\(m_{AB}\)[/tex] is the slope of [tex]\(\overleftrightarrow{AB}\)[/tex], then the slope [tex]\(m_{BC}\)[/tex] of [tex]\(\overleftrightarrow{BC}\)[/tex] is:
[tex]\[ m_{BC} = -\frac{1}{m_{AB}} = -\frac{1}{0.7142857142857143} \approx -1.4 \][/tex]
3. Find the y-intercept of the line [tex]\(\overleftrightarrow{BC}\)[/tex]:
Using the point-slope form of a line equation [tex]\(y - y_1 = m(x - x_1)\)[/tex], where [tex]\((x_1, y_1)\)[/tex] is the point [tex]\(B (4, 4)\)[/tex] and [tex]\(m\)[/tex] is the slope [tex]\(m_{BC}\)[/tex]:
[tex]\[ y - 4 = -1.4(x - 4) \][/tex]
Simplify and solve for [tex]\(y\)[/tex] to get the equation in slope-intercept form [tex]\(y = mx + c\)[/tex]:
[tex]\[ y - 4 = -1.4x + 5.6 \][/tex]
[tex]\[ y = -1.4x + 5.6 + 4 \][/tex]
[tex]\[ y = -1.4x + 9.6 \][/tex]
4. State the equation of line [tex]\(\overleftrightarrow{BC}\)[/tex]:
The equation of the line [tex]\(\overleftrightarrow{BC}\)[/tex] is:
[tex]\[ y = -1.4x + 9.6 \][/tex]
So, the correct equation of line [tex]\(\overleftrightarrow{BC}\)[/tex] is:
[tex]\[ y = -1.4x + 9.6 \][/tex]
1. Determine the slope of line [tex]\(\overleftrightarrow{AB}\)[/tex]:
The coordinates of points [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are given as [tex]\(A = (-3, -1)\)[/tex] and [tex]\(B = (4, 4)\)[/tex].
The formula to find 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 the coordinates of points [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:
[tex]\[ m_{AB} = \frac{4 - (-1)}{4 - (-3)} = \frac{4 + 1}{4 + 3} = \frac{5}{7} \approx 0.7142857142857143 \][/tex]
2. Calculate the slope of line [tex]\(\overleftrightarrow{BC}\)[/tex]:
Since [tex]\(\overleftrightarrow{BC}\)[/tex] is perpendicular to [tex]\(\overleftrightarrow{AB}\)[/tex] at point [tex]\(B\)[/tex], the slope of [tex]\(\overleftrightarrow{BC}\)[/tex] will be the negative reciprocal of the slope of [tex]\(\overleftrightarrow{AB}\)[/tex]. If [tex]\(m_{AB}\)[/tex] is the slope of [tex]\(\overleftrightarrow{AB}\)[/tex], then the slope [tex]\(m_{BC}\)[/tex] of [tex]\(\overleftrightarrow{BC}\)[/tex] is:
[tex]\[ m_{BC} = -\frac{1}{m_{AB}} = -\frac{1}{0.7142857142857143} \approx -1.4 \][/tex]
3. Find the y-intercept of the line [tex]\(\overleftrightarrow{BC}\)[/tex]:
Using the point-slope form of a line equation [tex]\(y - y_1 = m(x - x_1)\)[/tex], where [tex]\((x_1, y_1)\)[/tex] is the point [tex]\(B (4, 4)\)[/tex] and [tex]\(m\)[/tex] is the slope [tex]\(m_{BC}\)[/tex]:
[tex]\[ y - 4 = -1.4(x - 4) \][/tex]
Simplify and solve for [tex]\(y\)[/tex] to get the equation in slope-intercept form [tex]\(y = mx + c\)[/tex]:
[tex]\[ y - 4 = -1.4x + 5.6 \][/tex]
[tex]\[ y = -1.4x + 5.6 + 4 \][/tex]
[tex]\[ y = -1.4x + 9.6 \][/tex]
4. State the equation of line [tex]\(\overleftrightarrow{BC}\)[/tex]:
The equation of the line [tex]\(\overleftrightarrow{BC}\)[/tex] is:
[tex]\[ y = -1.4x + 9.6 \][/tex]
So, the correct equation of line [tex]\(\overleftrightarrow{BC}\)[/tex] is:
[tex]\[ y = -1.4x + 9.6 \][/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 visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Thank you for visiting Westonci.ca, your go-to source for reliable answers. Come back soon for more expert insights.
