Answer: 61/2
Step-by-step explanation:
To find the area of triangle ABC, we need to determine the coordinates of point C, which lies on the line [tex]\( M[/tex] that is perpendicular to line and intersects the line [tex]\( x = -1 \)[/tex]
The slope [tex]\( m \)[/tex] of a line passing through two points [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
So, for points [tex]\( A \)[/tex] and [tex]\( B \)[/tex], the slope [tex]\( m_L \)[/tex] will be:
[tex]\[ m_L = \frac{3 - 2}{5 - (-6)} \][/tex]
[tex]\[ m_L = \frac{1}{11} \][/tex]
Since line [tex]\( M \)[/tex] is perpendicular to line [tex]\( L \)[/tex], the slope of line [tex]\( M \)[/tex], [tex]\( m_M \)[/tex], is the negative reciprocal of [tex]\( m_L \)[/tex]:
[tex]\[ m_M = -\frac{1}{m_L} \][/tex]
[tex]\[ m_M = -\frac{1}{\frac{1}{11}} \][/tex]
[tex]\[ m_M = -11 \][/tex]
Now we need to find the midpoint of [tex]\( A \)[/tex] and [tex]\( B \)[/tex] to determine where line [tex]M \)[/tex] passes through. The midpoint [tex]\( D \)[/tex] has coordinates [tex]\( (x_D, y_D) \)[/tex] given by: [tex]\[ x_D = \frac{x_A + x_B}{2} \][/tex], and
[tex]\[ y_D = \frac{y_A + y_B}{2} \][/tex]
Let's calculate the midpoint D:
[tex]\[ x_D = \frac{-6 + 5}{2} \][/tex]
[tex]\[ x_D = \frac{-1}{2} \][/tex]
[tex]\[ y_D = \frac{2 + 3}{2} \][/tex]
[tex]\[ y_D = \frac{5}{2} \][/tex]
So, the midpoint D is [tex]\( \left(-\frac{1}{2}, \frac{5}{2}\right)[/tex]
Since line [tex]\( M \)[/tex] has a slope of =11 and passes through point [tex]\( D \)[/tex], we can write its equation using point-slope form:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
[tex]\[ y - \frac{5}{2} = -11\left(x + \frac{1}{2}\right) \][/tex]
To find point [tex]\( C \)[/tex], we need to substitute [tex]\( x = -1 \)[/tex] into the equation of line [tex]\( M \)[/tex] and solve for [tex]\( y \)[/tex]. The y-coordinate of point C, where line M intersects the line [tex]\( x = -1 \)[/tex], is [tex]\( y_C = 8 \)[/tex]. So the coordinates of point C are [tex]\( (-1, 8) \)[/tex]
Now that we have the coordinates of all three points A, B, and C:
- [tex]\( A(-6, 2) \)[/tex]
- [tex]\( B(5, 3) \)[/tex]
- [tex]\( C(-1, 8) \)[/tex]
We can calculate the area of triangle ABC as:
[tex]\[ \text{Area} = \frac{1}{2}|x_A(y_B - y_C) + x_B(y_C - y_A) + x_C(y_A - y_B)| \][/tex]
[tex]\[ \text{Area} = \frac{1}{2}|(-6)(3 - 8) + 5(8 - 2) + (-1)(2 - 3)| \][/tex]
[tex]\[ \text{Area} = \frac{1}{2}|30 + 30 + 1| \][/tex]
[tex]\[ \text{Area} = \frac{61}{2} \][/tex]
Therefore, the area of triangle ABC is 61/2 square units.
