In the `DeltaABC`, the coordinates of B are `(0, 0), AB=2, /_ABC=pi/3` and the middle point of BC has the coordinates (2,0). The centroid of the triangle is
A. `(1//2,sqrt3//2)`
B. `(5//3,1//sqrt3)`
C. `(4+sqrt3//3,1//3)`
D. none of these
Correct Answer - B
`(2,0)` is the midpoint of B(0,0) and C. Then C has coordinates (4,0). Also, A has coordinates `(0+2cospi//3,0+2sinpi//3)-=(1,sqrt(3))`. Then the centroid is `(5//3,1sqrt(3))`.
Correct Answer - `(2,5//3)`
As we know that the centroid of triangle ABC and that of the triangle formed by joining the middle points of the sides of triangle ABC are...
Correct Answer - `166.5` sq.unit
Let the given triangle be ABC.
centroid is `G(1,3)` and two vertices of triangle are `B(4,-8)` and `C(-9,7)`
`"Area of triangle GBC"=(1)/(2)||{:(1,,4),(4,,-8),(-9,,7),(1,,4):}||`
`=(1)/(2)|[-8-16+28-72-36-7]|`
`=555` sq.units.
Hence,...
Correct Answer - A::B::C::D
As H (orhtocenter), G (centroid), and C (circumcenter) are collinear we have
`|{:(4,,b,,1),(b,,2b-8,,1),(-4,,8,,1):}|=0`
or `|{:(4,,b,,1),(b-4,,b-8,,0),(-(b+4),,16-2b,,0):}|=0`
or `(b-4)(16-2b)+(b+4)(b-4)=0`
or `2(b-4)(8-b)+(b+4)+(b-8)=0`
or `(8-b)[2b-8)-(b+4)=0`
or `(8-b)(b-12)=0`
Hence `b=8 or 12`,...
Correct Answer - B
Let the coordinates of D be `(alpha,beta)`. Then,
`(alpha+1+3)/(3)=3` or `alpha=5`
and `(beta+2+4)/(3)=2` or `beta=0`
`therefore D-=(5,0)`
Taking `A(x_1,y_1),B(x_1,y_1)`, and `C(x_3,y_3)`, we have
`(x_1+x_2)/(2)=1`,
`(x_2+x_3)/(2) =5, (x_3+x_1)/(2)=3`...
Take A as the origin and last the postion vectors of point B and C be `vecb and vecc`, Respectively.
Therefore, the position vectors, of D, E and F are,...
Correct Answer - B
Circumradius of triangle ABC, R = 5
`:.` Circumradius of pedal triangle, `R_(1) = 5//2` and so on.
Now, `underset(i=1)overset(oo)sum R_(i) + R_(1) + R_(2) + R_(3)...