Coordinates of Centroid, Incentre and Excentre

Q. A (3, 2) and B (−2, 1) are two vertices of a triangle ABC whose centroid G has the coordinates $\left(\frac{5}{3},-\frac{1}{3}\right)$. Find the coordinates of the third vertex C of the triangle.

Q. Find the ratio in which the point P$\left(\frac{3}{4},\frac{5}{12}\right)$ divides the line segment joining the points A$\left(\frac{1}{2},\frac{3}{2}\right)$ and B$\left(2,-5\right)$. [CBSE 2015]

Q. Find the ratio in which the point P(−1, y) lying on the line segment joining A(−3, 10) and B(6 −8) divides it. Also find the value of y. [CBSE 2013]

Q. Two vertices of a triangle have coordinates (−8, 7) and (9, 4) . If the centroid of the triangle is at the origin, what are the coordinates of the third vertex?

Q.

The angle of the major sector and the corresponding minor sector of a circle are

1. Supplementary Angles

2. Complementary Angles

3. Conjugate Angles

4. Reflex Angles

Q.

The incenter of a triangle is :

1. Intersection of side bisectors of a triangle

2. Intersection of angle bisectors of a triangle

3. Intersection of altitudes of the triangle

4. Intersection of medians of the triangle

Q.

What will be the centroid of a triangle whose vertices are (2, 4), (6, 4) and (2, 0)?

1. $\frac{8}{3},\frac{5}{3}$
2. $\frac{7}{2},\frac{5}{2}$
3. $\frac{10}{3},\frac{8}{3}$
4. $\frac{4}{3},\frac{2}{3}$

Q. The points $A\left(x_1,y_1\right),B\left(x_2,y_2\right),C\left(x_3,y_3\right)$ are the vertices of $∆$ ABC .
(i) The median from A meets BC at D . Find the coordinates of the point D.
(ii) Find the coordinates of the point P on AD such that AP : PD = 2 : 1.
(iii) Find the points of coordinates Q and R on medians BE and CF respectively such that BQ : QE = 2 : 1 and CR : RF = 2 : 1.
(iv) What are the coordinates of the centropid of the triangle ABC ?

Q. Find the coordinates of the point which divides the join of (- 1, 7) and (4, - 3) in the ratio 2:3.

Q.

Find the centroids of the triangles whose vertices are given below.
(1) $(–7,6),(2,–2),(8,5)$

(2) $(3,–5),(4,3),(11,–4)$

(3) $(4,7),(8,4),(7,11)$

Q. Find the centroid of ∆ABC whose vertices are A(2, 2), B(−4, −4) and C(5, −8).

Q. In what ratio does the line x − y − 2 = 0 divide the line segment joining the points A(3, −1) and B(8, 9)?

Q.

The incenter of a triangle is given by:

1. $(x,y)\equiv (\frac{x_1+x_2+x_3}{a+b+c},\frac{y_1+y_2+y_3}{a+b+c})$

2. $(x,y)\equiv (\frac{a+b+c}{x_1+x_2+x_3},\frac{a+b+c}{y_1+y_2+y_3})$

3. $(x,y)\equiv (\frac{a{x}_1+b{x}_2+c{x}_3}{a+b+c},\frac{a{y}_1+b{y}_2+c{y}_3}{a+b+c})$

4. $(x,y)\equiv (a{x}_1+b{x}_2+c{x}_3,a{y}_1+b{y}_2+c{y}_3)$

Q.

The centroid of the triangle whose vertices are (1, 3) (2, 7) and (12, -16) is (5, 2)

1. True

2. False

Q.

The points (6, 6), (0, 6) and (6, 0) are the vertices of a right triangle as shown in the figure. Find the distance between its centroid and D (point of median of AB).

1. $\sqrt{3}\text{units}$

2. $\sqrt{6}\text{units}$

3. $\sqrt{2}\text{units}$

4. $\sqrt{5}\text{units}$

Q. Match the following columns:

Column I	Column II
(a) The coordinates of the point that divides the join of A(−1, 7) and B(4, −3) in the ratio 2 : 3 are .......	(p) (5, 6)
(b) Two vertices of ∆ABC are A(6, 4) and B(−2, 2) and its centroid is G(3, 4). The coordinates of C are .......	(q) $\sqrt{10}$
(c) If the points A(4, 3) and B(x, 5) lie on a circle with the centre O(2, 3), then x = ....... .	(r) (1, 3)
(d) If A(0, −1), B(2, 1) and C(0, 3) are the vertices of a ∆ABC, then the length of median AD is	(s) 2

Q. The ratio in which the point P(6, −6) divides the line segment joining A(1, 4) and B(9, −12) is

(a) 2 : 3
(b) 3 : 2
(c) 3 : 5
(d) 5 : 3