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 and incentre of the triangle whose vertices are

(2, 4), (6, 4), (2, 0)

.

Accepted Answer

Let the given vertices of the triangle be $A (2, 4), B (6, 4)$ and $C (2, 0)$

Now as we know the centroid of a triangle having vertices $(x_{1}, y_{1}), (x_{2}, y_{2})$ and $(x_{3}, y_{3})$ is given by $P (\frac{x_{1} + x_{2} + x_{3}}{3}, \frac{y_{1} + y_{2} + y_{3}}{3})$

$\Rightarrow$ The centroid of the given triangle $A B C$ is $P (\frac{2 + 6 + 2}{3}, \frac{4 + 4 + 0}{3})$

or $P (\frac{10}{3}, \frac{8}{3})$

Also we know that the incenter of a triangle having vertices $(x_{1}, y_{1}), (x_{2}, y_{2})$ and $(x_{3}, y_{3})$ and $a, b$ and $c$ be the length of the sides is given by $I (\frac{a x_{1} + b x_{2} + c x_{3}}{a + b + c}, \frac{a y_{1} + b y_{2} + c y_{3}}{a + b + c})$

Here $a = B C, b = A C$ and $c = A B$

Now as we know the distance between two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ is given by $\sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}$

$\Rightarrow A B = c = \sqrt{{(6 - 2)}^{2} + {(4 - 4)}^{2}} = 4$ units

and $B C = a = \sqrt{{(2 - 6)}^{2} + {(0 - 4)}^{2}} = \sqrt{16 + 16} = 4 \sqrt{2}$ units

and $C A = b = \sqrt{{(2 - 2)}^{2} + {(4 - 0)}^{2}} = 4$ units

So incenter of $A B C$ is $I (\frac{(4 \sqrt{2}) (2) + (4) (6) + (4) (2)}{4 \sqrt{2} + 4 + 4}, \frac{(4 \sqrt{2}) (4) + (4) (4) + (4) (0)}{4 \sqrt{2} + 4 + 4})$

$= I (\frac{8 \sqrt{2} + 24 + 8}{4 \sqrt{2} + 8}, \frac{16 \sqrt{2} + 16}{4 \sqrt{2} + 8})$

$= I (\frac{8 \sqrt{2} + 32}{4 \sqrt{2} + 8}, \frac{16 \sqrt{2} + 16}{4 \sqrt{2} + 8})$

$= I (\frac{2 \sqrt{2} + 8}{\sqrt{2} + 2}, \frac{4 \sqrt{2} + 4}{\sqrt{2} + 2})$

Find the centroid and incentre of the triangle whose vertices are $(2, 4), (6, 4), (2, 0)$ .

Important Questions on Cartesian System of Rectangular Coordinates

Learn and Prepare for any exam you want !

Find the centroid and incentre of the triangle whose vertices are 2,4,6,4,2,0.

Important Questions on Cartesian System of Rectangular Coordinates

Learn and Prepare for any exam you want !

Find the centroid and incentre of the triangle whose vertices are $(2, 4), (6, 4), (2, 0)$ .