Find the equations of the medians of the triangle with vertices (4,5), (2,-1), and (0,7).
this is easy using vectors. If we let the points be A,B,C, then if you extend the median from A to Ma to point D such that AMa = MaD you have a parallelogram with diagonal AD. AD = AB + AC = (-2,-6)+(-4,2) = (-6,-4)
So, the median AMa = A + (-3,-2) = (1,3)
The slope from A to Ma is 2/3
So the line through A with slope 2/3 is y-5 = 2/3 (x-4)
Now do the other two medians to points Mb and Mc
To find the equations of the medians of a triangle, we need to find the midpoints of each side of the triangle, and then find the equations of the lines passing through each midpoint and the opposite vertex.
Let's label the given points: A(4, 5), B(2, -1), and C(0, 7).
Step 1: Find the midpoints of the sides.
The midpoint formula is given by:
Midpoint = [(x1 + x2)/2, (y1 + y2)/2]
The midpoints of the sides AB, BC, and AC are as follows:
- Midpoint of AB: [(4 + 2)/2, (5 + (-1))/2] = (3, 2)
- Midpoint of BC: [(2 + 0)/2, (-1 + 7)/2] = (1, 3)
- Midpoint of AC: [(4 + 0)/2, (5 + 7)/2] = (2, 6)
Step 2: Find the equations of the lines passing through each midpoint and the opposite vertex.
To find the equation of a line, we need the slope and one point on the line. The slope can be found using the formula:
Slope = (y2 - y1)/(x2 - x1)
Let's find the equations of the lines passing through each midpoint and the opposite vertex:
Line passing through midpoint of AB (3, 2) and vertex C(0, 7):
- Slope = (7 - 2)/(0 - 3) = -5/3
- Using the point-slope form, we have:
y - 2 = (-5/3)(x - 3)
Simplifying it, we get:
y = (-5/3)x + 17/3
Line passing through midpoint of BC (1, 3) and vertex A(4, 5):
- Slope = (5 - 3)/(4 - 1) = 2/3
- Using the point-slope form, we have:
y - 3 = (2/3)(x - 1)
Simplifying it, we get:
y = (2/3)x + 7/3
Line passing through midpoint of AC (2, 6) and vertex B(2, -1):
- Slope = (-1 - 6)/(2 - 2) = -7/0 (undefined slope, since the denominator is zero)
- Since the slope is undefined, the equation of the line is x = 2
Therefore, the equations of the medians of the triangle are:
y = (-5/3)x + 17/3
y = (2/3)x + 7/3
x = 2