The mid-points of the sides of a triangle are | vedclass

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

Question

(N/A) Let the vertices of the triangle be $A(x_1, y_1, z_1)$,$B(x_2, y_2, z_2)$,and $C(x_3, y_3, z_3)$.Given mid-points are $D(5,7,11)$ on $BC$,$E(0,8

Vedclass · Accepted Answer

Vedclass · Answer

(N/A) Let the vertices of the triangle be $A(x_1, y_1, z_1)$,$B(x_2, y_2, z_2)$,and $C(x_3, y_3, z_3)$.
Given mid-points are $D(5,7,11)$ on $BC$,$E(0,8,5)$ on $AC$,and $F(2,3,-1)$ on $AB$.
Using the mid-point formula:
$1) \frac{x_2+x_3}{2} = 5, \frac{y_2+y_3}{2} = 7, \frac{z_2+z_3}{2} = 11 \Rightarrow x_2+x_3=10, y_2+y_3=14, z_2+z_3=22$
$2) \frac{x_1+x_3}{2} = 0, \frac{y_1+y_3}{2} = 8, \frac{z_1+z_3}{2} = 5 \Rightarrow x_1+x_3=0, y_1+y_3=16, z_1+z_3=10$
$3) \frac{x_1+x_2}{2} = 2, \frac{y_1+y_2}{2} = 3, \frac{z_1+z_2}{2} = -1 \Rightarrow x_1+x_2=4, y_1+y_2=6, z_1+z_2=-2$
Adding all equations for $x$:
$(x_2+x_3) + (x_1+x_3) + (x_1+x_2) = 10+0+4 = 14$ $\Rightarrow 2(x_1+x_2+x_3) = 14$ $\Rightarrow x_1+x_2+x_3 = 7$
$x_1 = (x_1+x_2+x_3) - (x_2+x_3) = 7-10 = -3$
$x_2 = (x_1+x_2+x_3) - (x_1+x_3) = 7-0 = 7$
$x_3 = (x_1+x_2+x_3) - (x_1+x_2) = 7-4 = 3$
Similarly for $y$:
$y_1+y_2+y_3 = (14+16+6)/2 = 18$
$y_1 = 18-14 = 4, y_2 = 18-16 = 2, y_3 = 18-6 = 12$
Similarly for $z$:
$z_1+z_2+z_3 = (22+10-2)/2 = 15$
$z_1 = 15-22 = -7, z_2 = 15-10 = 5, z_3 = 15-(-2) = 17$
Thus,the vertices are $A(-3, 4, -7)$,$B(7, 2, 5)$,and $C(3, 12, 17)$.

Column $I$	Column $II$
$(A)$ The centroid of the triangle formed by $(2, 3, -1)$,$(5, 6, 3)$,$(2, -3, 1)$ is	$(p)$ $(2, 2, 2)$
$(B)$ The circumcentre of the triangle formed by $(1, 2, 3)$,$(2, 3, 1)$,$(3, 1, 2)$ is	$(q)$ $(3, 1, 4)$
$(C)$ The orthocentre of the triangle formed by $(2, 1, 5)$,$(3, 2, 3)$,$(4, 0, 4)$ is	$(r)$ $(1, 1, 0)$
$(D)$ The incentre of the triangle formed by $(0, 0, 0)$,$(3, 0, 0)$,$(0, 4, 0)$ is	$(s)$ $(3, 2, 1)$

The mid-points of the sides of a triangle are $(5,7,11)$,$(0,8,5)$,and $(2,3,-1)$. Find its vertices.

Similar Questions

If $D(2, 1, 0)$,$E(2, 0, 0)$,and $F(0, 1, 0)$ are mid-points of the sides $BC$,$CA$,and $AB$ of $\triangle ABC$,respectively,then the centroid of $\triangle ABC$ is

If the midpoints of the sides $AB$,$BC$,and $CA$ of a triangle are respectively $D(1, 2, -3)$,$E(3, 0, 1)$,and $F(-1, 1, -4)$,then the centroid of the triangle $ADF$ is

$G(1,0,1)$ is the centroid of the triangle $ABC$. If $A=(1,-4,2)$ and $B=(3,1,0)$,then $AG^2+CG^2=$

If $A(-4, 5, p)$,$B(3, 1, 4)$,and $C(-2, 0, q)$ are the vertices of a triangle $ABC$ and $G(r, q, 1)$ is its centroid,then the value of $2p + q - r$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module