Write the equations for two medians in slope-intercept form and solve the system of equations.
(1, 53), see solution.
Practice makes perfect
We are given the coordinates of the triangle vertices A, B, and C. Let's plot these points on a coordinate plane and draw the triangle â–ł ABC.
Now, we can recall that a centroid of a triangle is a point of intersection of the triangle medians. Let's draw two medians and find their point of intersection.
First Median
In order to draw a median of a triangle, we need to choose a vertex and find a midpoint of the opposite to it side. Let's choose A. We can find the midpoint of BC by substituting the coordinates of B and C into the Midpoint Formula.
Now let's plot the midpoint of BC, which we can name N, and draw the median AN.
Let's now find the equation of AN. We will use the slope-intecept form. From the diagram, we can see that AN intersects the y-axis at 2, so the y-intercept of the line is 2.
b=2
To calculate the slope of AN, we can substitute the coordinates of A and N into the Slope Formula.
Now that we know both the slope and y-intercept of AN, we can write its equation in slope-intercept form.
y= mx+ b ⇒ y= - 13x+ 2
Second Median
Similarly, we can draw the second median from the vertex B. We can find the midpoint of the opposite side AC by substituting the coordinates of A and C into the Midpoint Formula.
Let's now plot the midpoint of AC, name it M, and draw the median BM.
We need to find the equation of BM in slope-intercept form. First, we can calculate the slope by substituting the coordinates of B and M into the Slope Formula.
So far the equation is the following.
y= mx+b ⇒ y= 10/3x+b
To find the y-intercept b, we will substitute point B(2,5) into the above equation and solve it for b.
The equation of BM in slope-intercept form is as follows.
y=10/3x+ b ⇒ y=10/3x+( - 5/3)
Find Centroid
We have found two equations of the medians of â–ł ABC. Using them, we can form a system of equations.
y=- 13x+2 y= 103x+(- 53)
If we solve it we will find the common solution for both equations. It is a point of intersection of these medians and the centroid that we were asked to find. Let's use the Substitution Method.
