At Westonci.ca, we connect you with the best answers from a community of experienced and knowledgeable individuals. Experience the convenience of getting reliable answers to your questions from a vast network of knowledgeable experts. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
To find the equation of the line passing through point \( C \) and perpendicular to line segment \( \overline{A B} \), follow these steps:
1. Find the slope of line segment \( \overline{A B} \):
The coordinates of point \( A \) are \( (2, 9) \) and those of point \( B \) are \( (8, 4) \).
The slope \( m_{AB} \) is calculated as:
[tex]\[ m_{AB} = \frac{B_y - A_y}{B_x - A_x} = \frac{4 - 9}{8 - 2} = \frac{-5}{6} \][/tex]
2. Find the slope of the line perpendicular to \( \overline{A B} \):
The slope \( m_{\perpendicular} \) of the perpendicular line is the negative reciprocal of \( m_{AB} \):
[tex]\[ m_{\perpendicular} = -\frac{1}{m_{AB}} = -\frac{1}{-\frac{5}{6}} = \frac{6}{5} = 1.2 \][/tex]
3. Use the point-slope form to write the equation of the line passing through \( C(-3, -2) \):
The point-slope form of a line is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where \( (x_1, y_1) \) is point \( C \) and \( m \) is the slope. So:
[tex]\[ y - (-2) = 1.2(x - (-3)) \][/tex]
4. Convert to slope-intercept form \( y = mx + b \):
Simplify and solve for \( y \):
[tex]\[ y + 2 = 1.2(x + 3) \][/tex]
[tex]\[ y + 2 = 1.2x + 3.6 \][/tex]
[tex]\[ y = 1.2x + 3.6 - 2 \][/tex]
[tex]\[ y = 1.2x + 1.6 \][/tex]
So, the complete equation of the line passing through point \( C \) and perpendicular to \( \overline{A B} \) is:
[tex]\[ y = 1.2x + 1.6 \][/tex]
Thus, the equation in the given format is:
[tex]\[ y = \boxed{1.2}x + \boxed{1.6} \][/tex]
1. Find the slope of line segment \( \overline{A B} \):
The coordinates of point \( A \) are \( (2, 9) \) and those of point \( B \) are \( (8, 4) \).
The slope \( m_{AB} \) is calculated as:
[tex]\[ m_{AB} = \frac{B_y - A_y}{B_x - A_x} = \frac{4 - 9}{8 - 2} = \frac{-5}{6} \][/tex]
2. Find the slope of the line perpendicular to \( \overline{A B} \):
The slope \( m_{\perpendicular} \) of the perpendicular line is the negative reciprocal of \( m_{AB} \):
[tex]\[ m_{\perpendicular} = -\frac{1}{m_{AB}} = -\frac{1}{-\frac{5}{6}} = \frac{6}{5} = 1.2 \][/tex]
3. Use the point-slope form to write the equation of the line passing through \( C(-3, -2) \):
The point-slope form of a line is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where \( (x_1, y_1) \) is point \( C \) and \( m \) is the slope. So:
[tex]\[ y - (-2) = 1.2(x - (-3)) \][/tex]
4. Convert to slope-intercept form \( y = mx + b \):
Simplify and solve for \( y \):
[tex]\[ y + 2 = 1.2(x + 3) \][/tex]
[tex]\[ y + 2 = 1.2x + 3.6 \][/tex]
[tex]\[ y = 1.2x + 3.6 - 2 \][/tex]
[tex]\[ y = 1.2x + 1.6 \][/tex]
So, the complete equation of the line passing through point \( C \) and perpendicular to \( \overline{A B} \) is:
[tex]\[ y = 1.2x + 1.6 \][/tex]
Thus, the equation in the given format is:
[tex]\[ y = \boxed{1.2}x + \boxed{1.6} \][/tex]
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.