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To determine the equation of the altitude from vertex B of triangle [tex]\( \triangle ABC \)[/tex] with vertices [tex]\( A(10,2) \)[/tex], [tex]\( B(4,-2) \)[/tex], and [tex]\( C(-4,6) \)[/tex], we will follow these steps:
1. Calculate the slope of segment [tex]\( AC \)[/tex]:
The slope [tex]\( m_{AC} \)[/tex] of line segment [tex]\( AC \)[/tex] is given by the formula:
[tex]\[ m_{AC} = \frac{y_C - y_A}{x_C - x_A} \][/tex]
Substitute the coordinates of points [tex]\( A \)[/tex] and [tex]\( C \)[/tex]:
[tex]\[ m_{AC} = \frac{6 - 2}{-4 - 10} = \frac{4}{-14} = -\frac{2}{7} \][/tex]
Hence, the slope of [tex]\( AC \)[/tex] is [tex]\[ -0.2857142857142857 \][/tex].
2. Determine the slope of the altitude from vertex [tex]\( B \)[/tex]:
The altitude from [tex]\( B \)[/tex] to [tex]\( AC \)[/tex] is perpendicular to [tex]\( AC \)[/tex]. The slope of a line perpendicular to another line is the negative reciprocal of the slope of the original line.
So, the slope of the altitude from [tex]\( B \)[/tex], [tex]\( m_{\text{altitude}} \)[/tex], is calculated as:
[tex]\[ m_{\text{altitude}} = -\frac{1}{m_{AC}} \][/tex]
Substitute the value of [tex]\( m_{AC} \)[/tex]:
[tex]\[ m_{\text{altitude}} = -\frac{1}{-\frac{2}{7}} = \frac{7}{2} = 3.5 \][/tex]
Hence, the slope of the altitude is [tex]\[ 3.5 \][/tex].
3. Find the equation of the line passing through [tex]\( B \)[/tex] with slope [tex]\( m_{\text{altitude}} \)[/tex]:
We use the point-slope form of the equation of a line, which is given by:
[tex]\[ y - y_1 = m (x - x_1) \][/tex]
Here, [tex]\( (x_1, y_1) \)[/tex] is point [tex]\( B(4, -2) \)[/tex] and [tex]\( m \)[/tex] is [tex]\( 3.5 \)[/tex]:
[tex]\[ y - (-2) = 3.5 (x - 4) \][/tex]
Simplifying this equation:
[tex]\[ y + 2 = 3.5x - 14 \][/tex]
4. Convert the equation to the general form [tex]\( Ax + By + C = 0 \)[/tex]:
Start with the simplified equation:
[tex]\[ y + 2 = 3.5x - 14 \][/tex]
Rearrange terms to get all terms to one side:
[tex]\[ y + 2 - 3.5x + 14 = 0 \][/tex]
[tex]\[ -3.5x + y + 16 = 0 \][/tex]
Thus, the equation of the altitude from vertex [tex]\( B \)[/tex] in general form is:
[tex]\[ -3.5x + y + 16 = 0 \][/tex]
So, we have successfully determined the equation of the altitude from vertex [tex]\( B \)[/tex]. The final altitude equation is:
[tex]\[ -3.5x + y + 16 = 0 \][/tex]
1. Calculate the slope of segment [tex]\( AC \)[/tex]:
The slope [tex]\( m_{AC} \)[/tex] of line segment [tex]\( AC \)[/tex] is given by the formula:
[tex]\[ m_{AC} = \frac{y_C - y_A}{x_C - x_A} \][/tex]
Substitute the coordinates of points [tex]\( A \)[/tex] and [tex]\( C \)[/tex]:
[tex]\[ m_{AC} = \frac{6 - 2}{-4 - 10} = \frac{4}{-14} = -\frac{2}{7} \][/tex]
Hence, the slope of [tex]\( AC \)[/tex] is [tex]\[ -0.2857142857142857 \][/tex].
2. Determine the slope of the altitude from vertex [tex]\( B \)[/tex]:
The altitude from [tex]\( B \)[/tex] to [tex]\( AC \)[/tex] is perpendicular to [tex]\( AC \)[/tex]. The slope of a line perpendicular to another line is the negative reciprocal of the slope of the original line.
So, the slope of the altitude from [tex]\( B \)[/tex], [tex]\( m_{\text{altitude}} \)[/tex], is calculated as:
[tex]\[ m_{\text{altitude}} = -\frac{1}{m_{AC}} \][/tex]
Substitute the value of [tex]\( m_{AC} \)[/tex]:
[tex]\[ m_{\text{altitude}} = -\frac{1}{-\frac{2}{7}} = \frac{7}{2} = 3.5 \][/tex]
Hence, the slope of the altitude is [tex]\[ 3.5 \][/tex].
3. Find the equation of the line passing through [tex]\( B \)[/tex] with slope [tex]\( m_{\text{altitude}} \)[/tex]:
We use the point-slope form of the equation of a line, which is given by:
[tex]\[ y - y_1 = m (x - x_1) \][/tex]
Here, [tex]\( (x_1, y_1) \)[/tex] is point [tex]\( B(4, -2) \)[/tex] and [tex]\( m \)[/tex] is [tex]\( 3.5 \)[/tex]:
[tex]\[ y - (-2) = 3.5 (x - 4) \][/tex]
Simplifying this equation:
[tex]\[ y + 2 = 3.5x - 14 \][/tex]
4. Convert the equation to the general form [tex]\( Ax + By + C = 0 \)[/tex]:
Start with the simplified equation:
[tex]\[ y + 2 = 3.5x - 14 \][/tex]
Rearrange terms to get all terms to one side:
[tex]\[ y + 2 - 3.5x + 14 = 0 \][/tex]
[tex]\[ -3.5x + y + 16 = 0 \][/tex]
Thus, the equation of the altitude from vertex [tex]\( B \)[/tex] in general form is:
[tex]\[ -3.5x + y + 16 = 0 \][/tex]
So, we have successfully determined the equation of the altitude from vertex [tex]\( B \)[/tex]. The final altitude equation is:
[tex]\[ -3.5x + y + 16 = 0 \][/tex]
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