Centroid of a Triangle

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.

In an equilateral $△ABC$ coordinates of the centroid of the triangle is ($1,2,3)$. Find the coordinates of the orthocenter of the triangle.

1. $(1,2,3)$

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

3. $(1,2,\infty )$

4. $(0,2,3)$

Q.

Which of the following statements are correct?

1. For triangle orthocenter, circumcenter and

centroid are collinear.

2. Centroid divides the line joining the circumcenter $\&$ orthocenter

in the ratio of $2:1$ i.e., $\frac{CS}{OC}=\frac{2}{1}$

Where C-Coordinate of centroid

O-Coordinate of orthocenter

S--Coordinate of circumcenter

1. Only 1

2. Only 2

3. Both 1 & 2

4. None of these

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

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

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

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

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. If Origin is the centroid of a triangle A(a, 1, 3), B(-2, b, -5) and C(4, 7, c), then values of a, b and c will be respectively.

1. -2, -8, 2
2. 2, 8, 2
3. -2, 8, 2
4. 2, 8, -2

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) be the origin, then the values of a, b, c are

1. – 2, –8, –2

2. 2, 8, –2

3. –2, –8, 2

4. 7, –1, 0

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