The centroid of a triangle $\mathrm{ABC}$ is at the point $(1{,}1{,}1)$. If the coordinates of $\mathrm{A}$ and $\mathrm{B}$ are $(3,-5{,}7)$ and $(-1{,}7,-6)$, respectively, find the coordinates of the point C.

Step 1: Recall the Centroid Formula

The centroid $G$ of a triangle with vertices $A(x_A, y_A, z_A)$, $B(x_B, y_B, z_B)$, and $C(x_C, y_C, z_C)$ 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 $x_C, y_C, z_C$ 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 ($x_C, y_C, z_C$).

Step 4: Solve for $x_C, y_C, z_C$

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 $x_C, y_C,$ and $z_C$, we get the coordinates of point C.

More from this chapter

• Find the equation of set of points P such that PA²+PB²=2 k², where A and B are the points (3,4,5) and (-1,3,-7), respectively.
• Show that the points A (1, 2, 3), B (-1, -2, -1), C (2, 3, 2) and D(4,7,6) are the vertices of a parallelogram ABCD, but it is not a rectangle.
• Are the points A (3,6,9), B (10,20,30) and C(25,-41,5), the vertices of a right angled triangle?
• Find the equation of the set of the points P such that its distances from the points A(3,4,-5) and B(-2,1,4) are equal.
• Show that the points P(-2,3,5), Q(1,2,3) and R(7,0,-1) are collinear.

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