Name: How to Find Locus of Any Point
Uploaded: 2019-07-19
Duration: 13 min 14 s
Description: In this video, we will learn about the locus of a point in 3D, which will help you to solve questions in exams based on this topic. Watch and learn.

Question

Find the centroid of the triangle $A B C$ whose vertices are $A (9, 2), B (1, 10)$ and $C (- 7, - 6)$ . Find the coordinates of the middle points of its sides and hence find the centroid of the triangle formed by joining these middle points. Do the two triangles have the same centroid?

Accepted Answer

The vertices of a triangle are $A (9, 2), B (1, 10)$ and $C (- 7, - 6)$

Here, $x_{1} = 9, x_{2} = 1, x_{3} = - 7$ and $y_{1} = 2, y_{2} = 10, y_{3} = - 6$

Let the coordinates of the centroid be $(x, y)$
So,Centroid of triangle $(x, y) = (\frac{x_{1} + x_{2} + x_{3}}{3}, \frac{y_{1} + y_{2} + y_{3}}{3})$

$= (\frac{9 + 1 + (- 7)}{3}, \frac{2 + 10 + (- 6)}{3})$ $= (1, 2)$

Hence, the centroid of a triangle is $(1, 2)$

Now,

Let $D, E$ and $F$ are the midpoints of the sides $B C, C A$ and $A B$ respectively.

The coordinates of $D$ are:
$D = [\frac{- 7 + 1}{2}, \frac{- 6 + 10}{2}]$

$D = (- 3, 2)$

The coordinates of $E$ are:

$E = [\frac{- 7 + 9}{2}, \frac{- 6 + 2}{2}]$

$E = (1, - 2)$
The coordinates of $F$ are:

$F = [\frac{1 + 9}{2}, \frac{10 + 2}{2}]$

$F = (5, 6)$

Now, we find the centroid of a triangle formed by joining these middle points $D, E$ , and $F$ as shown in figure

Let $P$ be the trisection point of the median $A D$ which is nearer to the opposite side $B C$

$∴ P$ divides $D A$ in the ratio $1 : 2$ internally

$∴ P = (\frac{1 (9) + 2 (- 3)}{1 + 2}, \frac{1 (2) + 2 (2)}{1 + 2})$

$= (\frac{9 - 6}{3}, \frac{2 + 4}{3})$ $= (1, 2)$

Let $Q$ be the trisection point of the median $B E$ which is nearer to the opposite side $C A$

$∴ Q$ divides $E B$ in the ratio $1 : 2$ internally

$∴ Q = (\frac{1 (1) + 2 (1)}{1 + 2}, \frac{1 (10) + 2 (- 2)}{1 + 2})$ $= (1, 2)$

Let $R$ be the trisection point of the median $C F$ which is nearer to the opposite side $A B$

$∴ R$ divides $F C$ in the ratio $1 : 2$ internally

$∴ R = (\frac{1 (- 7) + 2 (5)}{1 + 2}, \frac{1 (- 6) + 2 (6)}{1 + 2})$ $= (1, 2)$

Yes, the triangle has the same centroid, i.e. $(1, 2)$

Find the centroid of the triangle $A B C$ whose vertices are $A (9, 2), B (1, 10)$ and $C (- 7, - 6)$ . Find the coordinates of the middle points of its sides and hence find the centroid of the triangle formed by joining these middle points. Do the two triangles have the same centroid?

Important Questions on Cartesian System of Rectangular Coordinates

Learn and Prepare for any exam you want !

Find the centroid of the triangle ABC whose vertices are A (9,2), B(1,10) and C(-7,-6). Find the coordinates of the middle points of its sides and hence find the centroid of the triangle formed by joining these middle points. Do the two triangles have the same centroid?

Important Questions on Cartesian System of Rectangular Coordinates

Learn and Prepare for any exam you want !

Find the centroid of the triangle $A B C$ whose vertices are $A (9, 2), B (1, 10)$ and $C (- 7, - 6)$ . Find the coordinates of the middle points of its sides and hence find the centroid of the triangle formed by joining these middle points. Do the two triangles have the same centroid?