Given Trinagle ABC with A(-4, -2), B(4,4), and C(18,-8), answer the following questions.
1. Write the equation of the line containing the altitude that passes through B in standard form.
2. Write the equation of the line containing the median that passes through point C in slope-intercept form.
3. Write an equation for the line containing a perpendicular bisector of AC in point-slope form
To find the equations requested, we need to utilize the properties of altitudes, medians, and perpendicular bisectors.
Altitude:
The altitude of a triangle is a line segment drawn from a vertex of the triangle to the opposite side and is perpendicular to that side.
1. To find the equation of the line containing the altitude that passes through B, we need to find the slope of the line. The slope of an altitude is the negative reciprocal of the slope of the line containing the side it is perpendicular to.
The side opposite B is AC. The slope of line AC can be found using the slope formula:
slope = (y2 - y1) / (x2 - x1)
slope_AC = (-8 - (-2)) / (18 - (-4)) = -10 / 22 = -5/11
The slope of the altitude passing through B will be the negative reciprocal, which is 11/5.
Now, we can find the equation of the line using the point-slope form (y-y1) = m(x-x1) and then convert it to standard form (Ax + By = C).
Using point B (4,4) and slope 11/5:
(y - 4) = (11/5)(x - 4)
5(y - 4) = 11(x - 4)
5y - 20 = 11x - 44
11x - 5y = 24
So, the equation of the line containing the altitude that passes through B is 11x - 5y = 24 (in standard form).
Median:
The median of a triangle is a line segment drawn from a vertex to the midpoint of the opposite side.
2. To find the equation of the line containing the median that passes through point C, we need to find the coordinates of the midpoint of AB first.
The midpoint formula is:
midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)
Midpoint_AB = ((-4 + 4) / 2, (-2 + 4) / 2) = (0, 1)
Now we can find the equation of the line using the slope-intercept form y = mx + b.
Using the midpoint (0, 1) and point C (18, -8):
Slope = (y2 - y1) / (x2 - x1) = (-8 - 1) / (18 - 0) = -9 / 18 = -1 / 2
Plug in the slope and the coordinates of point C to the slope-intercept form:
y = (-1/2)x + b
Using point C (18, -8):
-8 = (-1/2) * 18 + b
-8 = -9 + b
b = 1
So, the equation of the line containing the median that passes through point C is y = (-1/2)x + 1 (in slope-intercept form).
Perpendicular Bisector:
The perpendicular bisector of a segment is a line that is perpendicular to the segment and passes through the midpoint of the segment.
3. To find the equation of the line containing the perpendicular bisector of AC, we need to find the coordinates of the midpoint of AC first.
Midpoint_AC = ((-4 + 18) / 2, (-2 - 8) / 2) = (7, -5)
Now we can find the slope of AC using the slope formula:
slope_AC = (-8 - (-2)) / (18 - (-4)) = -10 / 22 = -5/11
The slope of the perpendicular bisector will be the negative reciprocal, which is 11/5.
Using the point-slope form (y - y1) = m(x - x1) and the coordinates for the midpoint (7, -5):
(y - (-5)) = (11/5)(x - 7)
y + 5 = (11/5)x - (77/5)
y = (11/5)x - (77/5) - 5
y = (11/5)x - (77/5) - (25/5)
y = (11/5)x - (102/5)
So, the equation of the line containing the perpendicular bisector of AC is y = (11/5)x - (102/5) (in point-slope form).