Given ABC with A(-4,-2), B(4,4) and C(18,-8) answer the question
write the equation for line containing perpendicular bisector of AC in point slope form, show work.
AAAaannndd the bot gets it wrong yet again!
The midpoint of AC is (7,-5)
The slope of AC is -3/11 so the slope of the perp. is 11/3
y+5 = 11/3 (x-7)
The question is totally messed up by the bot ....
correction by a human:
midpoint of AC = (7,-5)
slope AC = (-8+2)/(18+4) = -6/22 = -3/11
slope of perpendicular bisector = 11/3
equation:
y + 5 = (11/3(x - 7)
To find the equation of the line containing the perpendicular bisector of AC in point-slope form, we first need to find the slope of AC and then determine the slope of the line perpendicular to AC.
Step 1: Find the midpoint of AC.
The midpoint of a line with endpoints (x₁, y₁) and (x₂, y₂) is given by the formula:
Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2)
So, the midpoint of AC is:
Midpoint = ((-4 + 18)/2, (-2 - 8)/2)
= (14/2, -10/2)
= (7, -5)
Step 2: Find the slope of AC.
The slope of a line with endpoints (x₁, y₁) and (x₂, y₂) is given by the formula:
Slope = (y₂ - y₁)/(x₂ - x₁)
Using points A(-4,-2) and C(18,-8), the slope of AC is:
Slope_AC = (-8 - (-2))/(18 - (-4))
= (-8 + 2)/(18 + 4)
= -6/22
= -3/11
Step 3: Find the slope of the line perpendicular to AC.
Any two lines that are perpendicular to each other have slopes that are negative reciprocals of each other. So, if the slope of AC is m, then the slope of the line perpendicular to AC is -1/m.
Therefore, the slope of the perpendicular line is:
Slope_perpendicular = -1/(-3/11)
= 11/3
Step 4: Use the point-slope form equation.
The equation of a line in point-slope form is given by:
y - y₁ = m(x - x₁)
Using the midpoint (7, -5) and the slope of the perpendicular line 11/3, the equation becomes:
y - (-5) = (11/3)(x - 7)
Simplifying the equation gives:
y + 5 = (11/3)(x - 7)
This is the equation for the line containing the perpendicular bisector of AC in point-slope form.
The equation of the perpendicular bisector of AC is the line that passes through the midpoint of AC and is perpendicular to AC.
The midpoint of AC is (7, -3).
The slope of AC is -6/14.
The slope of the perpendicular bisector of AC is 14/6.
Therefore, the equation of the perpendicular bisector of AC in point slope form is y - (-3) = (14/6)(x - 7).