# Video: Finding the Equation of a Plane Cutting the Coordinate Axes given the Intersection Point of the Medians of a Triangle

Find the equation of the plane cutting the coordinate axes at π΄, π΅, and πΆ, given that the intersection point of the medians of β³π΄π΅πΆ is (π, π, π).

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### Video Transcript

Find the equation of the plane cutting the coordinate axes at π΄, π΅, and πΆ, given that the intersection point of the medians of triangle π΄π΅πΆ is π, π, π.

As we get started, we can sketch in the segment of our plane that intersects the π₯-, π¦-, and π§-axes. Weβre told that the π₯-value of the intercept of our plane with the π₯-axis is capital π΄ so that the coordinates of this point are capital π΄, zero, zero. The π¦-value of the intersection point along the π¦-axis is capital π΅. So this point has coordinates zero, capital π΅, zero and similarly for the π§-axis which has coordinates zero, zero, capital πΆ.

Now, weβre then told something interesting about this triangle π΄π΅πΆ. Weβre told that the coordinates of the intersection point of the medians of our triangle are π, π, π. 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, weβre told, has coordinates π, π, π. Based on all this, we want to find the equation of this plane. There are many different forms in which a planeβs equation can be expressed. But here, since weβre given the intercept points between our plane and the coordinate axes, weβll aim to write the equation of our plane in whatβs called intercept form.

This form is written in terms of the π₯-, π¦-, and π§-coordinates of the points of intersection on these respective axes, but our challenge is that we donβt know the values of capital π΄, π΅, and πΆ. We do, however, know the coordinates of the point at the centroid of π΄, π΅, and πΆ. To give the equation of our plane in terms of the known values π, π, and π, weβll recognize the fact that the coordinates of the centroid of a triangle are equal to the average values of the π₯-, π¦-, and π§-coordinates of its vertices. In other words, this value π is equal to the average of capital π΄, zero, and zero, the π₯-values of the three vertices. Likewise, π is equal to the average value of zero, capital π΅, and zero and so on.

We can write this out as follows: π is equal to π΄ plus zero plus zero over three. π equals zero plus π΅ plus zero over three. π equals zero plus zero plus πΆ over three. These equations imply that capital π΄ equals three π, capital π΅ equals three π, and capital πΆ equals three π. Knowing this, weβre now ready to write the equation of our plane in intercept form. We have π₯ divided by π΄ or three π plus π¦ divided by π΅ or three π plus π§ divided by πΆ or three π 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 π₯ over π plus π¦ over π plus π§ over π equals three.