ABC has vertices A(0, 6), B(3, 3), and C(6, 0). Write the equation of the line containing the altitude AR
y = -x
y = x
y = -x + 1
y= x - 1
PLEASE SAY THE CORRECT ANSWER AND WHY
y = -x + 6
Explanation: The equation of the line containing the altitude AR can be found by using the slope-intercept form of a line. The slope of the line is -1, since it is perpendicular to the line connecting A and B. The y-intercept is 6, since the altitude passes through the point A(0, 6). Therefore, the equation of the line containing the altitude AR is y = -x + 6.
A, B, and C lie on the same straight line, no triangle ABC is possible
proof:=
slope AB = -1
slope BC = -1 , they are linked at B, so ABC forms a line.
The "solution" provided by the AI is bogus and nonsense
To find the equation of the line containing the altitude AR of triangle ABC, we first need to find the slope of the line segment BC, which is perpendicular to the altitude.
The slope of line segment BC can be calculated using the formula:
slope = (change in y) / (change in x)
slope = (0 - 3) / (6 - 3)
slope = -3/3
slope = -1
The altitude, which is perpendicular to line segment BC, will have a slope that is the negative reciprocal of -1.
The negative reciprocal of -1 is +1, therefore the slope of the altitude is 1.
We can now use the slope-intercept form of the equation (y = mx + b) to find the equation of the line containing the altitude.
Since the point A(0, 6) lies on the altitude, we can substitute the coordinates (0, 6) into the equation:
6 = 1(0) + b
Simplifying this equation, we find:
b = 6
Therefore, the equation of the line containing the altitude AR is:
y = x + 6
So, the correct answer is y = x + 6.
To find the equation of the line containing the altitude AR in triangle ABC, we need to first determine the coordinates of point R.
An altitude of a triangle is a line segment drawn from a vertex of the triangle perpendicular to the opposite side. To find the coordinates of point R, we need to find the equation of the line containing the side BC (the opposite side of vertex A) and then calculate the intersection point of that line with the line passing through A and perpendicular to BC.
Let's start by finding the equation of the line containing side BC. We can use the slope-intercept form of a linear equation, y = mx + b, where m is the slope of the line and b is the y-intercept.
The slope of BC can be found by using the formula:
m = (y2 - y1) / (x2 - x1)
m = (3 - 0) / (3 - 6)
m = 3 / -3
m = -1
Since vertex B(3, 3) lies on the line, we can substitute the coordinates into the equation to find the y-intercept (b):
3 = -1 * 3 + b
3 = -3 + b
b = 6
Therefore, the equation of the line containing side BC is y = -x + 6.
Next, we need to find the equation of the line passing through A(0, 6) and perpendicular to BC. The slope of a line perpendicular to BC is the negative reciprocal of the slope of BC. So, the slope of the perpendicular line is 1.
Using the point-slope form of a linear equation, y - y1 = m(x - x1), where m is the slope and (x1, y1) are the coordinates of a point on the line, we can substitute the values to find the equation of the line containing the altitude AR:
y - 6 = 1(x - 0)
y - 6 = x
Therefore, the equation of the line containing the altitude AR in triangle ABC is y = x.
Hence, the correct answer is "y = x" because it represents the equation of the line containing the altitude AR.