Find the co-ordinates of the points which trisect the line segment joining the points $P(4,2,-6)$ and $Q(10,-16,6)$.

**OR**

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 $c$

Question

Find the co-ordinates of the points which trisect the line segment joining the points $P(4,2,-6)$ and $Q(10,-16,6)$.

**OR**

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 $c$.

Accepted Answer

>

> [!TLDR]
> The trisection points are $(6,-4,-2)$ and $(8,-10,2)$.
>
> For $P(4,2,-6)$ and $Q(10,-16,6)$, the two trisection points divide $PQ$ in ratios $1:2$ and $2:1$.
>
> **First point $R_1$ (ratio $1:2$ from $P$):**
> $$R_1 = \left(\dfrac{1(10)+2(4)}{3},\ \dfrac{1(-16)+2(2)}{3},\ \dfrac{1(6)+2(-6)}{3}\right) = \left(\dfrac{18}3,\dfrac{-12}3,\dfrac{-6}3\right)=(6,-4,-2).$$
>
> **Second point $R_2$ (ratio $2:1$ from $P$):** …

Plane	Equation	Description
$xy$ -plane	$z = 0$	the floor — all points with zero height
$yz$ -plane	$x = 0$	one wall — all points with zero $x$
$zx$ -plane	$y = 0$	the other wall — all points with zero $y$

Q.Find the co-ordinates of the points which trisect the line segment joining the points $P(4,2,-6)$ and $Q(10,-16,6)$ . OR 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 $c$ .

Concept understanding — 3D Coordinate Geometry

3D Coordinate Geometry

The Intuition: Three Numbers, One Point

The Precise Statement

How to Read a 3D Point

The Three Coordinate Planes

Distance Between Two Points

Section Formula (Internal Division)

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