The Centroid

Q.

The point of intersection of medians is called ___.

1. centroid

2. orthocentre

3. incentre

4. circumcentre

Q.

If the coordinates of the middle point of the portion of a line intercepted between the coordinate axes is $(3,2)$, then the equation of the line will be:

1. $2x+3y=12$

2. $3x+2y=12$

3. $4x-3y=6$

4. $5x-2y=10$

Q.

The locus of the midpoint of the line joining focus and any point on the parabola ${\mathrm{y}}^2=4\mathrm{ax}$ is a parabola with the equation of directrix as

1. $\mathrm{x}+\mathrm{a}=0$

2. $2\mathrm{x}+\mathrm{a}=0$

3. $\mathrm{x}=0$

4. $\mathrm{x}=\frac{\mathrm{a}}{2}$

Q.

The in-centre of a triangle with vertices $\left(7,1\right)$, $\left(-1,5\right)$and $\left(3+2\sqrt{3},3+4\sqrt{3}\right)$ is

1. $\left(3+\frac{2}{\sqrt{3}},3+\frac{4}{\sqrt{3}}\right)$

2. $\left(1+\frac{2}{3\sqrt{3}},1+\frac{4}{3\sqrt{3}}\right)$

3. $\left(7,1\right)$

4. None of these

Q.

A Stick Of Length $10Units$ Rests Against The Floor And A Wall Of A Room. If The Stick Begins To Slide On The Floor Then The Locus Of Its Middle Point Is

Q.

If AM and GM of $x$ and $y$ are in the ratio $p:q$, then $x:y$ is

1. $p–\surd (p^2+q^2):p+\surd (p^2+q^2)$

2. $p+\surd (p^2–q^2):p–\surd (p^2–q^2)$

3. $p:q$

4. None of these

Q.

In triangle $PQR$, $PQ=13,QR=14,andPR=15$.

Let $\mathrm{M}$ be the midpoint of $\mathrm{QR}$.

Find $\mathrm{PM}.$

Q. The point of concurrence of all 3 medians in a triangle is known as

1. Othrocentre
2. Centre
3. Centroid
4. Altitude

Q.

Point of intersection of all the angle bisector in a triangle is known as __________ .

1. Circumcentre

2. Centroid

3. Centre

4. Incentre

Q. A function satisfies the equation $\text{y}={\text{3x}}^2-5\text{x}+12$
Which of the following mapping diagram is true for the function?

1. 
2. 
3. 
4. 

Q.

Both Arjun and Rahul went for jogging the next day and stopped at a place where kids were playing on a slide as shown below:

Two slides which are directed in the two opposite directions are combined with the vertical stairs in the midway between the two sliders. The lengths of both the slides are same.

Two kids Mia and Anu reached down after sliding. How much distance Mia has to walk from $A$ to reach Anu who is at $B,$ if $AC=5m$ and width of the stairs $=1m?$

1. 5 m

2. 10 m

3. 11 m

4. 6 m

Q. The centroid divides each median of a triangle in the ratio ______.

Q. A variable plane which remains at a constant distance $3p$ from the origin cuts the coordinate axes at $A,B,C$. Show that the locus of the centroid of triangle $ABC$ is $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=\frac{9}{p^2}$.

Q.

Following are the measures of certain sides and angles of triangle. You have to identify which cannot be constructed and say why it cannot be constructed. Now, construct the triangle if possible.

In $△\mathrm{PQR},\mathrm{PQ}=2\mathrm{cm},\mathrm{QR}=4\mathrm{cm},\mathrm{PR}=2\mathrm{cm}$

Q. $D,E,F$ are the mid points of $BC,CA,andAB$ of $\Delta ABC$. If $AD$ and $BE$ intersect in $G$, then $AG+BG+CG$ is equal to

1. $AD=BE=CF$
2. $\frac{2}{3}(AD+BE+CF)$
3. $\frac{3}{2}(AD+BE+CF)$
4. $\frac{1}{3}(AD+BE+CF)$

Q. What is the minimum number of Medians that has to intersect in order to get the centroid in any triangle ?

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

Q. The incentre of a triangle coincides with its circumcentre, orthocentre and centroid in case of ___________.

1. An isosceles triangle
2. A right angled triangle
3. An equilateral triangle
4. A scalene triangle

Q. In $\Delta ABC$, if $A(1,-6),B(-5,2)$ and the centroid is $G(-2,1)$, then Co-ordinates of vertex $C$ are

1. $(-2,7)$
2. $(-2,1)$
3. $(-2,6)$
4. $(3,2)$

Q.

Suppose $ABC$ be a triangle , whose centroid is $G$. If the coordinates of $B,CandG$ are $(-2,0),(3,1),(1,5)$, then the equations of the sides $AB$ and $AC$ are respectively.

1. $7x-2y+14=0,13x+y-40=0$

2. $7x+2y+14=0,x+13y-40=0$

3. $7x-2y-14=0,13x+13y-40=0$

4. none of these

Q. If two vertices of a triangle are $(3,7)$ and $(-1,5)$ and its centroid is $(1,3)$. Find the coordinates of the third vertex.

1. $(-1,3)$
2. $(1,-3)$
3. $(1,3)$
4. None of these

Q. $A(1,2,3),B(2,-1,6),C(3,2,-3)$ are vertices of $△ABC$. Find the Centroid of Triangle

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

Q. If a straight line be such that algebraic sum of the perpendiculars drawn from the points $A_1,A_2$ _____, $A_n$ is zero, then prove that the straight line passes through a fixed point whose co-ordinate is the centroid $G$ of the given points.

Q.

In a circular table cover of radius 12 cm, a design is formed leaving an equilateral triangle ABC in the middle as shown in figure. Find the area of the design. [4 MARKS]

Q. $ABC$ is an isosceles triangle with $AB=AC=5$ and $BC=6$. If $G$ is the centroid of $△ABC$, then $AG$ is equal to

1. $\frac{1}{3}$
2. $\frac{2}{3}$
3. $\frac{4}{3}$
4. $\frac{8}{3}$

Q. If G is the centroid and AD be a median with length $12$ cm of $\Delta ABC$, then the value of AG is

1. $4$ cm
2. $8$cm
3. $10$ cm
4. $6$ cm

Q. Let $O(0,0),P(3,4),Q(6,0)$ be the vetices of the triangle $OPQ$. The point $R$ inside the triangle $OPQ$ is such that the triangles $OPR,PQR,OQR$ are of equal area. The co-ordinates of $R$ are

1. $(\frac{4}{3},3)$
2. $(3,\frac{4}{3})$
3. $(3,\frac{2}{3})$
4. $(\frac{4}{3},\frac{2}{3})$

Q. If orthocenter and circumcenter of a triangle are respectively $(1,1)$ and $(3,2)$ then the co ordinates of its centroid are

1. $(\frac{7}{3},\frac{5}{3})$
2. $(7,5)$
3. $(\frac{5}{3},\frac{7}{3})$
4. None of these

Q. Centroid of a triangle, the equation of whose sides are $12x^2-20xy+7y^2=0$ and $2x-3y+4=0$ is

1. $(2,2)$
2. $(\frac{8}{3},\frac{8}{3})$
3. $(1,1)$
4. $(\frac{4}{3},\frac{4}{3})$
   

Q. Two vertices of a triangle are the points $(1,4),(7,2)$ and centroid is the point $(5,3)$. Find the coordinates of the third vertex.

1. $(2,5)$
2. $(7,2)$
3. $(3,7)$
4. $(7,3)$

Q. If $G$ be the centroid of a triangle $ABC$ and $P$ be any other point in the plane, then $P{A}^2+P{B}^2+P{C}^2=G{A}^2+G{B}^2+G{C}^2+4G{P}^2$.

1. True
2. False