Determining the Coordinates of the Third Vertex in a Triangle Given the Centroid and Two Vertices
We know that the centroid (G) of a triangle with vertices A, B, and C can be found using the formula:
<p>\( G(\frac{x_A + x_B + x_C}{3},\frac{y_A + y_B + y_C}{3},\frac{z_A + z_B + z_C}{3}) \)</p>
Given that the centroid is at (1,1,1) and the coordinates of points A and B are A(3, −5,7) and B(−1,7,−6) respectively, we can plug these into the formula and solve for the coordinates of point C (x_C, y_C, z_C).
<p>\( 1 = \frac{3 + (-1) + x_C}{3} \)</p>
<p>\( 1 = \frac{2 + x_C}{3} \)</p>
<p>\( 3 = 2 + x_C \)</p>
<p>\( x_C = 1 \)</p>
<p>\( 1 = \frac{-5 + 7 + y_C}{3} \)</p>
<p>\( 1 = \frac{2 + y_C}{3} \)</p>
<p>\( 3 = 2 + y_C \)</p>
<p>\( y_C = 1 \)</p>
<p>\( 1 = \frac{7 + (-6) + z_C}{3} \)</p>
<p>\( 1 = \frac{1 + z_C}{3} \)</p>
<p>\( 3 = 1 + z_C \)</p>
<p>\( z_C = 2 \)</p>
Therefore, the coordinates of point C are (1, 1, 2).
