given triangle ABC with A(-3,2), B(-1,-4), and C(4,1) write the equation for the line containing altitude LINE AR in standard form.
To find the equation of the line containing altitude LINE AR, we need to find the coordinates of point R, which is the foot of the altitude from point A.
First, let's find the slope of the line BC. The slope of a line passing through points (x₁, y₁) and (x₂, y₂) is given by the formula:
m = (y₂ - y₁) / (x₂ - x₁)
So, the slope of line BC is:
m₁ = (-4 - 1) / (-1 - 4) = -5 / (-5) = 1
Since the altitude is perpendicular to line BC, the slope of the altitude is the negative reciprocal of the slope of line BC. The negative reciprocal of 1 is -1. Therefore, the slope of the altitude is -1.
Now, let's find the midpoint of line BC, which is also the coordinates of point R. The midpoint formula is:
x = (x₁ + x₂) / 2
y = (y₁ + y₂) / 2
Using the coordinates of points B and C, we can find the midpoint:
x = (-1 + 4) / 2 = 3/2
y = (-4 + 1) / 2 = -3/2
So, the coordinates of point R are (3/2, -3/2).
Now, we have the slope (-1) and a point (3/2, -3/2) on the line containing altitude LINE AR.
Using the point-slope form of a linear equation:
y - y₁ = m(x - x₁)
where m is the slope and (x₁, y₁) is a point on the line, we can substitute the values:
y - (-3/2) = -1(x - 3/2)
y + 3/2 = -x + 3/2
y = -x + 3/2 - 3/2
y = -x
Rearranging the equation in standard form:
x + y = 0
Therefore, the equation for the line containing altitude LINE AR in standard form is x + y = 0.