15. The vertices of a triangle are (−4,2), (0,6), and (7, −2) as shown on the following grid. The equation of the median from vertex is = 2.
a) Find the equation of the median from the vertex
First, we need to find the coordinates of the midpoint of the side opposite the vertex. Let's call this midpoint point M. We can find the coordinates of M by using the midpoint formula:
M = ((-4 + 7)/2, (2 - 2)/2) = (1.5, 0)
Next, we need to find the slope of the line connecting the vertex and the midpoint. We can use the slope formula:
m = (y2 - y1)/(x2 - x1)
where (x1, y1) are the coordinates of the vertex and (x2, y2) are the coordinates of the midpoint. Let's choose the vertex (-4, 2) and the midpoint (1.5, 0):
m = (0 - 2)/(1.5 - (-4)) = -0.25
Now we have the slope of the line connecting the vertex and the midpoint, but we need the slope of the median, which is perpendicular to this line. We can use the fact that the product of the slopes of two perpendicular lines is -1:
m1 * m2 = -1
where m1 is the slope of the line we just found and m2 is the slope of the median. Solving for m2, we get:
m2 = -1/m1 = -1/(-0.25) = 4
Now we have the slope of the median, and we also know a point on the median (the midpoint we found earlier, (1.5, 0)). We can use the point-slope form of a line to find the equation of the median:
y - y1 = m(x - x1)
where m is the slope we just found and (x1, y1) is the point we chose. Plugging in our values, we get:
y - 0 = 4(x - 1.5)
Simplifying, we get:
y = 4x - 6
So the equation of the median from the vertex (-4, 2) is y = 4x - 6.