The centroid of a triangle ABC is at the point (1,1,1). If the coordinates of A and B are (3,−5,7) and (−1,7,−6), respectively, find the coordinates of the point C.
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Step-by-Step Solution
Step 1: Recall the Centroid Formula
The centroid G of a triangle with vertices A(xA,yA,zA), B(xB,yB,zB), and C(xC,yC,zC) is given by the average of the coordinates of its vertices. This formula helps us relate the known centroid and vertex coordinates to the unknown coordinates of point C.
Step 2: Substitute Known Values
We are given the coordinates of the centroid G(1,1,1), point A(3,−5,7), and point B(−1,7,−6). We substitute these values into the centroid formula, leaving xC,yC,zC as unknowns.
Step 3: Formulate Equations for Each Coordinate
By simplifying the numerators and equating the corresponding coordinates of the centroid, we obtain three separate linear equations, one for each coordinate (xC,yC,zC).
Step 4: Solve for xC,yC,zC
We solve each linear equation for its respective unknown. For each equation, we multiply both sides by 3 and then isolate the variable to find the coordinates of point C.
Step 5: State the Coordinates of C
Combining the calculated values for xC,yC, and zC, we get the coordinates of point C.