If a student is given the 3 vertices (coordinates) of a triangle, they will be asked to find the centroid of the triangle. The definition of a centroid of a triangle is intersection of the medians of the triangle. There are 2 ways to find the coordinates of the centroid, with one being easier than another, but the students should be aware of the 2 different ways to solve this type of problem. In our example, we will be looking for the centroid given the vertices A (1, 2), B (3, 4), C (5, 0).
Solution #1: use the formula
Coordinate of centroid is [(x1 + x2 + x3)/3, (y1 + y2 + y3)/3]Applying the formula of the centroid, you will find the coordinate of the centroid to be
[(1 + 3 + 5)/3, (2 + 4 + 0)/3] = (9/3, 6/3) = (3,2)
Solution #2:
Find the mid points of 2 of the sides of the triangles. Using the 2 vertices across from the 2 mid points, you can find the equation of the line. Once you know the equations of the 2 lines, you can find the intersection of the 2 lines (which effectively gives you the x, y coordinate).
In our example, the mid point D between the line segment AB is [(1 + 3)/2, (2 + 4)/2] = (2, 3). Now we can use point D (2, 3) and vertex C (5, 0) to find the equation of the line CD. Slope of CD = (0 - 3)/(5 - 2) = -3/3 = -1. To find the equation of the line CD, we will need to use the slope intercept form of the equation:
y = mx + b
y = -1x + b (plug in either point C or point D)
0 = -1(5) + b
b = 5
line CD: y = -1x + 5
The mid point E between the line segment BC is [(3 + 5)/2, (4 + 0)/2] = (4, 2). Now we can use the point E (4, 2) and vertex A (1, 2) to find the equation of line AE. Slope of AE can be found with (2 - 2)/(4 - 1) = 0. Using the slope intercept form, we can find
y = mx + b (we can use either point A or point E)
2 = 0(4) + b
b = 2
line AE: y = 2
Now you have found 2 lines (AE and CD), and knowing that they intercept, we should be able to find the intersecting points.
Line AE: y = 2
Line CD: y = -1x + 5
Solving 2 linear equations, you will find the coordinates of the intersecting points, which is also the centroid of the triangle.
2 = -1x + 5
x = 3
So the intersection point is (3, 2), which is the same answer as solution #1.
If you have any question regarding this type of problems, please feel free to reach out to me or any of the instructors in my center.
Michael Huang
Center Director
Mathnasium of Glen Rock/Ridgewood
T: 201-444-8020
E: glenrock@mathnasium.com
www.mathnasium.com/glenrock
