The vertices of a triangle are A(13,13), B(9,3), and C(-1,1).
Determine the equation of the median AM, where M is the midpoint of BC.
Midpoint M of BC = ((9-1)/2,(3+1)/2)
=(4,2)
slope AM = (13-2)/(13-4) = 11/9
So the equation of the median is
11x - 9y + c = 0, (knowing that the slope of Ax + By + C = 0 is -A/B)
substitute one of the points, say (4,2)
11(4) - 9(2) + C = 0
C=-26
the equation of the median is 11x - 9y - 26 = 0
I use this method of finding the equation of a straight line if the slope is a fraction.
It gives the equation very quickly in the general form without any fractions.
To determine the equation of the median AM, we first need to find the coordinates of the midpoint M of BC.
The coordinates of point B are (9,3) and the coordinates of point C are (-1,1). To find the midpoint, we use the midpoint formula:
Midpoint M = ((x1 + x2)/2, (y1 + y2)/2)
Substituting the coordinates of points B and C into the formula, we get:
M = ((9 - 1)/2, (3 + 1)/2)
= (8/2, 4/2)
= (4, 2)
Now that we have the midpoint M, we can find the slope of the median AM. The slope of a line passing through two points (x1, y1) and (x2, y2) is given by the formula:
Slope = (y2 - y1)/(x2 - x1)
Substituting the coordinates of points A(13, 13) and M(4, 2) into the formula, we get:
Slope AM = (13 - 2)/(13 - 4)
= 11/9
We now have the slope of the median AM, but we need to determine the equation of the median. The equation of a straight line in general form is Ax + By + C = 0, where A, B, and C are constants.
Since the slope of the median AM is 11/9, we can write the equation as:
11x - 9y + C = 0
To find the value of C, we substitute the coordinates of one of the points on the line into the equation. Let's use the coordinates of point M(4, 2):
11(4) - 9(2) + C = 0
44 - 18 + C = 0
26 + C = 0
C = -26
Therefore, the equation of the median AM is:
11x - 9y - 26 = 0.