Centroid of a Triangle

Q. The plane $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=k$, meets the coordinate axes at A, B, C such that the centroid of the triangle ABC is the point (a, b, c). Then k is

1. 3
2. 2
3. 1
4. 5

Q. In a three dimensional co - ordinate system P, Q and R are images of a point A(a, b, c) in the xy the yz and the zx planes respectively. If G is the centroid of triangle PQR then area of triangle AOG is (O is the origin)

1. $0$
2. $a^2+b^2+c^2$
3. $\frac{2}{3}(a^2+b^2+c^2)$
4. $\frac{1}{2}(a^2+b^2+c^2)$

Q. If the centroid of triangle whose vertices are (a, 1, 3), (–2, b, –5) and (4, 7, c) is origin, then the value of c – a – b is _____
___

Q.

A plane meets the coordinate axes at points A, B, C and $(\alpha ,\beta ,\gamma )$ is the centroid of the triangle ABC. Then the equation of the plane is

1. $\alpha x+\beta y+\gamma z=1$

2. $\frac{x}{\alpha }+\frac{y}{\beta }+\frac{z}{\gamma }=3$

3. $\frac{x}{\alpha }+\frac{y}{\beta }+\frac{z}{\gamma }=1$

4. $\frac{3x}{\alpha }+\frac{3y}{\beta }+\frac{3z}{\gamma }=1$

Q.

A variable plane which remains at a constant distance 3p from the origin cuts the co-ordinate axes at A, B, C. The locus of the centroid of triangle ABC is

1. $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{3p}$

2. $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=\frac{1}{9p^2}$

3. $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{p}$

4. $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=\frac{1}{p^2}$

Q.

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.

Q.

If a vertex of a triangle is (1, 1) and the mid-points of the two sides through this vertex are (-1, 2) and (3, 2), then the centroid of the triangle is _____.

1. $(-1,\frac{7}{3})$

2. $(-\frac{1}{3},\frac{7}{3})$

3. $(1,\frac{7}{3})$

4. $(\frac{1}{3},\frac{7}{3})$

Q. If the centroid of the triangle formed by $(7,x),(y,-6)$ and $(9,10)$ is at $(6,3)$, then $(x,y)$ is :

1. (5, 2)
2. (3, 1)
3. (4, 0)
4. (1, 3)

Q. If the centroid of the triangle formed by the points (a, b), (b, c) and (c, a) is at the origin, then $a^3+b^3+c^3$ is equal to

1. abc
2. 0
3. a+b+c
4. 3abc

Q.

Two vertices of a triangle are (3, –5) and (–7, 4). If its centroid is (2, –1). Find the third vertex.

1. (10, -3)

2. (12, -2)

3. (10, -2)

4. (9, -2)