Centroid of a Triangle

Q.

An equilateral triangle of side $9\mathrm{cm}$ is inscribed in a circle then the radius of the circle is?

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.

The vertices of a triangle are $(6,0),(0,6)$ and $(6,6)$. The distance between its circumcenter and centroid is

1. $2$

2. $\sqrt{2}$

3. $1$

4. $2\sqrt{2}$

Q.

If G be the centroid of a triangle ABC, prove that :

$A{B}^2+B{C}^2+C{A}^2=3(G{A}^2+G{B}^2+G{C}^2)$

Q.

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

1. $(1,7/3)$

2. $(1/3,7/3)$

3. $(-1,7/3)$

4. $(-1,7/3)$

Q. If in an equilateral triangle ABC, AD is median and in triangle ABD, AE is median, then AB square: AE square will be_________ .

Q. Which of the following statements is / are correct statements?
1. Centroid of a triangle is the point of concurrency of medians.
2. Incentre of a triangle is the point of concurrency of perpendicular bisectors of the sides of the triangle.
3. Circumcentre of a triangle is the point of concurrency of internal bisectors of the angles of the triangle.
4. Orthocentre of a triangle is the point of concurrency of altitudes of the triangle drawn from one vertex to opposite side.
5. A triangle can have only one excentre.

1. only 1, 2, 3, 4

2. only 1, 4

3. only 1, 4, 5

4. All 1, 2, 3, 4, 5

Q. In $△$ABC , G(-4, -7) is the centroid. If the coordinates of A and B are (-14, -19) and (3, 5) respectively, then find the co-ordinates of C.

1. (-2, -7)
2. (1, 7)
3. (-1, -7)
4. (-1, 7)

Q.

Let $A(4,2),B(6,5)$and $C(1,4)$ be the vertices of $∆ABC$.

(i) The median from $A$ meets $BC$ at $D$. Find the coordinates of point $D$.

(ii) Find the coordinates of the point $P$ on $AD$ such that $AP:PD=2:1$.

(iii) Find the coordinates of point $Q$ and $R$ on medians $BE$ and $CF$ respectively such that $BQ:QE=2:1$ and $CR:RF=2:1$.

(iv) What do you observe? [Note : The point which is common to all the three medians is called the centroid and this point divides each median in the ratio $2:1$].

(v) If $A(x_1,y_1),B(x_2,y_2)$ and $C(x_3,y_3)$ are the vertices of triangle $ABC$, find the coordinates of the centroid of the triangle.

Q. In triangle ABC, the median from A is given perpendicular to the median from B. If BC=7cm and AC=6cm, then the length of AB is?

Q. Question 7 (v)
Let A (4, 2), B (6, 5) and C (1, 4) be the vertices of triangle ABC.
(e) If $A(x_1,y_1),B(x_2,y_2)andC(x_3,y_3)$ are the vertices of triangle ABC, find the coordinates of the centroid of the triangle.

Q.

Two vertices of a triangle are $(-1,4)$ and $(5,2)$. If the centroid is $(0,-3)$ then find the third vertex.

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.

The in-centre of a triangle with vertices $(1,\sqrt{3})$, $(0,0)$, and $(2,0)$ is

1. $\left(1,\frac{1}{\sqrt{3}}\right)$

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

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

4. None of these

Q. In a triangle, centroid is the point of intersection of

1. altitudes.
2. perpendicular bisector.
3. medians.
4. angular bisector.

Q.

If $G$ be the centroid of a triangle $ABC$, prove that $A{B}^2+B{C}^2+A{C}^2=3\left(G{A}^2+G{B}^2+G{C}^2\right)$.

Q. A(h, -6) , B(2, 3) and C(-6, k) are co-ordinates of vertices of a triangle whose centroid is G(1, 5). What is the values of (h - k)?

1. -11

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

Q.

Which of the points are points of trisection of A(2, -2) and B(-7, 4)?

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. 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. In PQR, Q = 90, seg QM is the median. PQ² + QR² = 169. Draw a circumcircle of PQR.

Q. Find the centroid of the triangle whose vertices are $A(4,-6),B(3,-2)$ and $C(5,2)$

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

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

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

Q.

If the coordinates of the mid points of the sides of a triangle are (1, 1), (2, – 3) and (3, 4) Find its centroid. [4 MARKS]

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)

Q. Find the centroid of triangle $A(2,2),B(1,7),C(3,0)$

Q. Find the centroid of the triangle formed by the point (a, b), (b, c), (c, a) at the origin then find the value of a³ + b³ + c^3.

Q. $ABC$ is a triangle in a plane with vertices $A(2,3,5),B(-1,3,2)$ and $C(\lambda ,5,\mu )$. If the median through A is equally inclined to the coordinate axes, then the value of $({\lambda }^3+{\mu }^3+5)$ is:

1. $1348$
2. $1077$
3. $1130$
4. $676$

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