# Centroid

## Trending Questions

**Q.**In a triangle ABC, coordinates of A are (1, 2) and the equations of the medians through B and C are respectively, x+y=5 and x=4. Then area of △ABC (in sq. units) is :

- 12
- 4
- 5
- 9

**Q.**

Let $a,b,c$ be in arithmetic progression. Let the centroid of the triangle with vertices $(a,c),(2,b)$ and $(a,b)$ be $\left(\frac{10}{3},\frac{7}{3}\right)$. If$\mathrm{\xce\pm},\mathrm{\xce\xb2}$ are the roots of the equation $a{x}^{2}+bx+1=0$, then the value of ${\mathrm{\xce\pm}}^{2}+{\mathrm{\xce\xb2}}^{2}-\mathrm{\xce\pm}\mathrm{\xce\xb2}$ is:

$\frac{71}{256}$

$-\frac{69}{256}$

$\frac{69}{256}$

$-\frac{71}{256}$

**Q.**A triangle has a vertex at (1, 2) and the mid points of the two sides through it are (–1, 1) and (2, 3). Then the centroid of this triangle is :

- (1, 73)
- (13, 53)
- (13, 1)
- (13, 2)

**Q.**Let α, β, γ be the roots of the equation x3−12x2+44x−48=0. If the coordinates of the vertices of a triangle are A(α, 1α), B(β, 1β) and C(γ, 1γ), then the centroid of the △ABC is

- (1136, 4)
- (4, 1136)
- (3, 511)
- (511, 3)

**Q.**

Let $A\left(2,-3\right)$, $B\left(-2,1\right)$, be the vertices of a $\xe2\u02c6\u2020ABC$. If the centroid of this triangle moves on the line $2x+3y=1$, then the locus of the vertex is the line.

$3x-2y=3$

$2x+3y=9$

$2x-3y=7$

$3x+2y=5$

**Q.**

The centroid of an equilateral triangle is (0, 0). If two vertices of the triangle lie on x + y = 2√2 , then one of them will have its coordinates as

- (√2+√6, √2−√6)
- (√2+√3, √2−√3)
- (√2+√5, √2−√5)
- (√2+√6, √2+√6)

**Q.**The locus of centroid of a triangle whose vertices are (acost, asint), (bsint, −bcost) and (1, 0), where t is a parameter, is

- (3x−1)2+(3y)2=a2−b2
- (3x−1)2+(3y)2=a2+b2
- (3x+1)2+(3y)2=a2+b2
- (3x+1)2+(3y)2=a2−b2

**Q.**A point P moves on the line 2x−3y+4=0. If Q(1, 4) and R(3, −2) are fixed points, then the locus of the centroid of ΔPQR is a line :

- parallel to x-axis
- with slope 32
- parallel to y-axis
- with slope 23

**Q.**Let O(0, 0), P(3, 4), Q(6, 0) be the vertices of the triangle OPQ. The point R inside the tringle OPQ is such that the triangles OPR, PQR, OQR are of equal area. The coordinates of R are

- (43, 3)
- (3, 23)
- (3, 43)
- (43, 23)

**Q.**

The centroid of a triangle is (2, 7) and two of its vertices are (4, 8) and (−2, 6). The third vertex is

(7, 4)

(7, 7)

(0, 0)

(4, 7)

**Q.**If the centroid and a vertex of an equilateral triangle are (2, 3) and (4, 3) respectively, then the other two vertices of the triangle are

- (−1, 3+√3) and (−1, 3−√3)
- (1, 3+√3) and (1, 3−√3)
- (2, 3+√3) and (2, 3−√3)
- (1, 3+2√3) and (1, 3−2√3)

**Q.**

If the points (1, -1), (2, -1) and (4, -3) are the mid-points of the sides of a triangle, then write the coordinates of its centroid.

**Q.**

Find the direction cosines of the sides of the triangle whose vertices are (3, 5, -4), (-1, 1, 2) and (-5, -5, -2).

**Q.**If C be the centroid of the triangle having vertices (3, −1), (1, 3) and (2, 4). Let P be the point of intersection of the lines x+3y−1=0 and 3x−y+1=0, then the line passing through the points C and P also passes through the point :

- (−9, −7)
- (−9, −6)
- (9, 7)
- (7, 6)

**Q.**A quadratic function y=px2+qx+r, where p, q and r are distinct real numbers satisfies the below conditions :

(1) The graph of y passes through the point (1, 6).

(2) 1p, 1q, 1r form an arithmetic progression.

(3) p, r, q are in geometric progression.

Which of the following options is/are correct?

- Sum of the roots of y=0 is −4.
- Product of the roots of y=0 is −2.
- There exist 2 real and distinct roots of y=0
- There exist 2 real and equal roots of y=0.

**Q.**From any point P lying in first quadrant on the ellipse x225+y216=1 PN is drawn perpendicular to the major axis such that N lies on major axis. Now PN is produced to the point Q such that NQ equals to PS, where S is a focus. The point Q lies on which of the following lines

- 2y-3x-25=0
- 3x+5y+25=0
- 2x-5y+25=0
- 2x-5y-25=0

**Q.**For △ABC whose vertices are given by A(3, 6), B(−4, 7) and C(10, −7), if A1, B1, C1 represent the mid points of sides opposite to vertices A, B, C respectively, of △ABC and A2, B2, C2 represent the mid points of sides opposite to vertices A1, B1, C1 respectively, of △A1B1C1 and so on, then which of the following is/are correct?

- The centroid of △A2B2C2 is (3, 2)
- The centroid of △A3B3C3 is (3, 2)
- Area (△A2B2C2)Area (△ABC)=116
- Area (△A3B3C3)Area (△ABC)=116

**Q.**Let p, q and r be the roots of the equation y3−3y2+6y+1=0. If the vertices of a triangle are (pq, 1pq), (qr, 1qr) and (rp, 1rp), then the coordinates of its centroid are

- (1, 2)
- (2, 3)
- (1, −1)
- (2, −1)

**Q.**If (53, 3) is the centroid of a triangle and its two vertices are (0, 1) and (2, 3), then the coordinates of the third vertex are

- (1, 2)
- (2, 3)
- (3, 4)
- (3, 5)

**Q.**For the triangle formed by the point (4, 8), (2, 8), (4, 12)

- Orthocentre ≡(5, 7)
- Orthocentre ≡(4, 8)
- Circumcentre ≡(3, 10)
- Centroid ≡(103, 283)

**Q.**

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

2√2

√2

2

1

**Q.**

How do you find the terminal point $p(x,y)$ on the unit circle determined by the given value of $t=-\frac{3\mathrm{\xcf\u20ac}}{4}$ ?

**Q.**For â–³ABC whose vertices are given by A(3, 6), B(âˆ’4, 7) and C(10, âˆ’7), if A1, B1, C1 represent the mid points of sides opposite to vertices A, B, C respectively, of â–³ABC and A2, B2, C2 represent the mid points of sides opposite to vertices A1, B1, C1 respectively, of â–³A1B1C1 and so on, then which of the following is/are correct?

- Area (â–³A2B2C2)Area (â–³ABC)=116
- Area (â–³A3B3C3)Area (â–³ABC)=116
- The centroid of â–³A3B3C3 is (3, 2)
- The centroid of â–³A2B2C2 is (3, 2)

**Q.**Consider a △ABC, whose vertices are A(−2, 3, −4), B(4, 2, 1) and C(1, 1, 0), then the distance (in units) of its centroid from origin is

- 2
- √6
- 4
- 2√5

**Q.**The locus of centroid of a triangle whose vertices are (acost, asint), (bsint, −bcost) and (1, 0), where t is a parameter, is

- (3x−1)2+(3y)2=a2−b2
- (3x−1)2+(3y)2=a2+b2
- (3x+1)2+(3y)2=a2+b2
- (3x+1)2+(3y)2=a2−b2

**Q.**43. Consider 3 non collinear points A, B, C with coordinates (0, 6), (5, 5) and (-1, 1) respectively.Equation of a line tangent to the circle circumscribing the triangle ABC and passing through the origin is where x < y is 1. 11 2. 7 3. 9 4. 13

**Q.**Let p, q and r be the roots of the equation y3−3y2+6y+1=0. If the vertices of a triangle are (pq, 1pq), (qr, 1qr) and (rp, 1rp), then the coordinates of its centroid are

- (1, 2)
- (2, −1)
- (1, −1)
- (2, 3)

**Q.**

Find the coordinates of the incentre and centroid of the triangle whose sides have the equations 3x−4y=0, 12y+5x=0 and y−15=0.

**Q.**The area of an equilateral triangle whose two vertices are (1, 0) and (3, 0) and third vertex lying in the first quadrant is-

- √3/4
- √3/2
- √3
- None of these

**Q.**One vertex of the equilateral triangle with centriod at origin and one side as x+y−2=0 is

- (−2, −2)
- (2, −2)
- (−2, 2)
- (2, 2)