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If orthocentre and circumcentre of triangle are respectively (1, 1) and (3, 2) then the co-ordinates of its centroid are:

A) \( \Large \left(\frac{7}{3},\ \frac{5}{3}\right) \)
B) \( \Large \left(\frac{5}{3},\ \frac{7}{3}\right) \)
C) (7, 5)
D) none of these

Correct Answer:

A) \( \Large \left(\frac{7}{3},\ \frac{5}{3}\right) \)

Description for Correct answer:

As we know that orthocentre, centroid and cricumcentre are collinear and centroid divides the line segment joining ortho centre and circumcentre in the ratio 2 : 1. If the points of orthocentre and circumcentre are \( \Large \left(1,\ 1\right)\ and\ \left(3,\ 2\right) \) respectively the co-ordinate of centroid is:

\( \Large \left(\frac{2.3+1.1}{2+1},\ \frac{2.2+1.1}{2+1}\right) = \left(\frac{7}{3},\ \frac{5}{3}\right) \)

Part of solved Rectangular and Cartesian products questions and answers : >> Elementary Mathematics >> Rectangular and Cartesian products

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Similar Questions

1). If
\( \begin{vmatrix}
x_{1} & y_{1} & 1 \\
x_{2} & y_{2} & 1 \\
x_{3} & y_{3} & 1
\end{vmatrix} = 0 \) , then the points \( \Large \left(x_{1},\ y_{1}\right),\ \left(x_{2},\ y_{2}\right)\ and\ \left(x_{3},\ y_{3}\right) \) are:

A). Vertices of an equilateral triangle

B). Vertices of a right angled triangle

C). Vertices of an isosceles triangle

D). none of these

2). If two vertices of an equilateral triangle are \( \Large \left(0,\ 0\right)\ and\ \left(3,\ 3\sqrt{3}\right) \) then the third vertex is:

A). \( \Large \left(3,\ -3\right) \)

B). \( \Large \left(-3,\ 3\right) \)

C). \( \Large \left(-3,\ 3\sqrt{3}\right) \)

D). none of these

3). If origin is shifted to \( \Large \left(7,\ -4\right) \) then point \( \Large \left(4,\ 5\right) \) shifted to

A). \( \Large \left(-3,\ 9\right) \)

B). \( \Large \left(3,\ 9\right) \)

C). \( \Large \left(11,\ 1\right) \)

D). none of these

4). The feet of the perpendicular drawn from P to the sides of a triangle ABC are collinear, then P is:

A). circumcentre of triangle ABC

B). lies on the circumcircle of triangle ABC

C). excentre of triangle ABC

D). none of these

5). The points \( \Large \left(k,\ 2-2k\right) \), \( \Large \left(-k+1,\ 2k\right) \), \( \Large \left(-4-k,\ 6-2k\right) \) are collinear then k is equal to:

A). 2, 3

B). 1, 0

C). \( \Large \frac{1}{2},\ 1 \)

D). 1, 2

6). Let AB is divided internally and externally at P and Q in the same ratio. Then AP, AB, AQ are in

A). AP

B). GP

C). HP

D). none of these

7). If O be the origin and if \( \Large P_{1} \left(x_{1},\ y_{1}\right)\ and\ P_{2} \left(x_{2},\ y_{2}\right) \) be two points, then \( \Large \left(OP_{1} \parallel \ OP_{2} \right) \cos \left( \angle P_{1}\ OP_{2}\right) \) is equal to:

A). \( \Large x_{1}y_{2}+x_{2}y_{1} \)

B). \( \Large \left(x^{2}_{1}+x^{2}_{2}+y^{2}_{2}\right) \)

C). \( \Large \left(x_{1}-x_{2}\right)^{2}+ \left(y_{1}-y_{2}\right)^{2} \)

D). \( \Large x_{1}x_{2}+y_{1}y_{2} \)

8). The median BE and AD of triangle with vertices \( \Large A \left(0,\ b\right), B \left(0,\ 0\right)\ and\ C \left(a,\ 0\right) \) are perpendicular to each other if;

A). \( \Large a=\frac{b}{2} \)

B). \( \Large b=\frac{a}{2} \)

C). ab = 1

D). \( \Large a\ =\ \pm \sqrt{2}b \)

9). ABC is a triangle with vertices A = (-1, 4), B = (6, -2) and C = (-2, 4). D, E and F are points which divide each AB, BC and CA respectively in the ratio 3 : 1 internally. Then centroid of the triangle DEF is:

A). \( \Large \left(3,\ 6\right) \)

B). \( \Large \left(1,\ 2\right) \)

C). \( \Large \left(4,\ 8\right) \)

D). \( \Large \left(-3,\ 6\right) \)

10). If \( \Large A \left(x_{1},\ y_{1}\right), B \left(x_{2},\ y_{2}\right)\ and\ C \left(x_{3},\ y_{3}\right) \) are the vertices of a triangle, then the excentre with respect to B is:

A). \( \Large \frac{ \left(ax_{1}-bx_{2}+cx_{3}\right) }{a-b+c},\ \frac{ay_{1}-by_{2}+cy_{3}}{a-b+c} \)

B). \( \Large \frac{ \left(ax_{1}+bx_{2}-cx_{3}\right) }{a+b-c},\ \frac{ay_{1}+by_{2}-cy_{3}}{a-b-c} \)

C). \( \Large \frac{ \left(ax_{1}-bx_{2}-cx_{3}\right) }{a-b-c},\ \frac{ay_{1}-by_{2}-cy_{3}}{a-b-c} \)

D). none of these