Find the equation of the plane cutting the coordinate axes at 𝐴, 𝐵, and 𝐶, given that the intersection point of the medians of triangle 𝐴𝐵𝐶 is 𝑙, 𝑚, 𝑛.
As we get started, we can sketch in the segment of our plane that intersects the 𝑥-, 𝑦-, and 𝑧-axes. We’re told that the 𝑥-value of the intercept of our plane with the 𝑥-axis is capital 𝐴 so that the coordinates of this point are capital 𝐴, zero, zero. The 𝑦-value of the intersection point along the 𝑦-axis is capital 𝐵. So this point has coordinates zero, capital 𝐵, zero and similarly for the 𝑧-axis which has coordinates zero, zero, capital 𝐶.
Now, we’re then told something interesting about this triangle 𝐴𝐵𝐶. We’re told that the coordinates of the intersection point of the medians of our triangle are 𝑙, 𝑚, 𝑛. The three medians of our triangle are lines that go from the corners to the midpoint of the opposite side of the triangle. As we see, they all meet at a point. And that point, we’re told, has coordinates 𝑙, 𝑚, 𝑛. Based on all this, we want to find the equation of this plane. There are many different forms in which a plane’s equation can be expressed. But here, since we’re given the intercept points between our plane and the coordinate axes, we’ll aim to write the equation of our plane in what’s called intercept form.
This form is written in terms of the 𝑥-, 𝑦-, and 𝑧-coordinates of the points of intersection on these respective axes, but our challenge is that we don’t know the values of capital 𝐴, 𝐵, and 𝐶. We do, however, know the coordinates of the point at the centroid of 𝐴, 𝐵, and 𝐶. To give the equation of our plane in terms of the known values 𝑙, 𝑚, and 𝑛, we’ll recognize the fact that the coordinates of the centroid of a triangle are equal to the average values of the 𝑥-, 𝑦-, and 𝑧-coordinates of its vertices. In other words, this value 𝑙 is equal to the average of capital 𝐴, zero, and zero, the 𝑥-values of the three vertices. Likewise, 𝑚 is equal to the average value of zero, capital 𝐵, and zero and so on.
We can write this out as follows: 𝑙 is equal to 𝐴 plus zero plus zero over three. 𝑚 equals zero plus 𝐵 plus zero over three. 𝑛 equals zero plus zero plus 𝐶 over three. These equations imply that capital 𝐴 equals three 𝑙, capital 𝐵 equals three 𝑚, and capital 𝐶 equals three 𝑛. Knowing this, we’re now ready to write the equation of our plane in intercept form. We have 𝑥 divided by 𝐴 or three 𝑙 plus 𝑦 divided by 𝐵 or three 𝑚 plus 𝑧 divided by 𝐶 or three 𝑛 equaling one. If we multiply both sides of this equation by three, we get this final result. The plane that meets the conditions described here has the equation 𝑥 over 𝑙 plus 𝑦 over 𝑚 plus 𝑧 over 𝑛 equals three.