A uniform triangular lamina has vertices 𝐴 equals seven, one; 𝐵 equals nine, three; and 𝐶 equals eight, five. Find the coordinates of its center of mass.
Okay, in this exercise, we have a uniform triangular lamina, where a lamina is a thin sheet of material that has mass. We know the coordinates of the three vertices of this triangle. And if we were to sketch them in on an 𝑥𝑦-coordinate plane, vertex 𝐴 would be here at seven, one; vertex 𝐵 would be here at nine, three; and vertex 𝐶 would be here at eight, five. Our triangle then looks like this, but actually we don’t need to sketch it in order to answer this question. That’s because for any uniform triangular lamina, like we have here, its center of mass is located at the average 𝑥- and 𝑦-values of its vertices.
In other words, if we solve for the average 𝑥-value of its vertices, that equals the 𝑥-coordinate of this triangle’s center of mass. And the same is true for the average 𝑦-value of the vertices. Since we’re given the coordinates of our three vertices, we can calculate these average values. The average 𝑥-value of the vertices is seven plus nine plus eight divided by three, which equals eight. Similarly, the average 𝑦-value of the vertices is one plus three plus five divided by three. That equals three. And therefore, the center-of-mass coordinates of this triangle are eight, three. And note that this method works for any uniform triangular lamina.