If the origin is the centroid of the triangle | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) The coordinates of the centroid of a triangle with vertices $(x_1, y_1, z_1)$,$(x_2, y_2, z_2)$,and $(x_3, y_3, z_3)$ are given by $\left(\frac{x_

Vedclass · Accepted Answer

$a = -2, b = -\frac{16}{3}, c = 2$

Vedclass · Answer

(A) The coordinates of the centroid of a triangle with vertices $(x_1, y_1, z_1)$,$(x_2, y_2, z_2)$,and $(x_3, y_3, z_3)$ are given by $\left(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}, \frac{z_1+z_2+z_3}{3}\right)$.Given the vertices $P(2a, 2, 6)$,$Q(-4, 3b, -10)$,and $R(8, 14, 2c)$,the centroid is:$\left(\frac{2a - 4 + 8}{3}, \frac{2 + 3b + 14}{3}, \frac{6 - 10 + 2c}{3}\right) = \left(\frac{2a + 4}{3}, \frac{3b + 16}{3}, \frac{2c - 4}{3}\right)$.Since the origin $(0, 0, 0)$ is the centroid,we equate the coordinates:$\frac{2a + 4}{3} = 0$ $\Rightarrow 2a = -4$ $\Rightarrow a = -2$.$\frac{3b + 16}{3} = 0$ $\Rightarrow 3b = -16$ $\Rightarrow b = -\frac{16}{3}$.$\frac{2c - 4}{3} = 0$ $\Rightarrow 2c = 4$ $\Rightarrow c = 2$.Thus,the values are $a = -2, b = -\frac{16}{3}, c = 2$.

If the origin is the centroid of the triangle $PQR$ with vertices $P(2a, 2, 6)$,$Q(-4, 3b, -10)$,and $R(8, 14, 2c)$,then find the values of $a, b$,and $c$.

Similar Questions

Three vertices of a parallelogram are $(1, 3)$,$(2, 0)$,and $(5, 1)$. Find its fourth vertex.

The perimeter of the triangle with vertices at $(1,0,0), (0,1,0)$ and $(0,0,1)$ is

If the points $A(1, 3, 5)$, $B(2, 4, 6)$, and $C(4, 5, k)$ form a right-angled triangle, then the number of possible values of $k$ is:

If the circumcenter of the triangle formed by the points $(1,2,3), (3,-1,5)$ and $(4,0,-3)$ is $(\alpha, \beta, \gamma)$, then $|\alpha|+|\beta|=$

If $G(4, 3, 3)$ is the centroid of the triangle $ABC$ whose vertices are $A(a, 3, 1)$,$B(4, 5, b)$,and $C(6, c, 5)$,then the values of $a, b, c$ are:

Vedclass Test Series

Exam Paper Generator

Online Exam Module