find the coordinates of the incentre and centroid of the triangle whose sides have the equation 3x -4y=0, 12y+5x=0, y-15=0.

also please tell me what is incentre, circumcentre, orthocentre and centroid in a triangle

Question

find the coordinates of the incentre and centroid of the
triangle whose sides have the equation 3x -4y=0, 12y+5x=0,
y-15=0
also please tell me what is incentre, circumcentre, orthocentre
and centroid in a triangle

Ayush Jha · Accepted Answer

Centroid of Triangle-

The centroid of a triangle is the point of concurrency of the medians. The centroid G of the triangle ABC, divides the median AD, in the ratio of 2 : 1.

Incentre of Triangle-

The incentre ‘I’ of a triangle is the point of concurrency of the bisectors of the angles of the triangle.

Circum Centre of Triangle-

This the point of concurrency of the perpendicular bisectors of the sides of the triangle. This is also the centre of the circle, passing through the vertices of the given triangle.

Orthocentre of Triangle-

This is the point of concurrency of the altitudes of the triangle.

We have to find the co-ordinates of the centroid and the incentre of the triangle which is formed by the 3 lines whose equations are-

3x-4y=0

12y+5x=0

y-15=0

Solve the 3 linear equation taking 2 at a time to get the co-ordinates of the vertices of the triangle as-

A(0,0) B(20,15) C(-36,15)

Now use distance formula to calculate the sides of the triangle as,

$A B (c) = \sqrt{{(20 - 0)}^{2} + {((15 - 0))}^{2}} = 25 B C (a) = \sqrt{{(20 + 36)}^{2} + {((15 - 15))}^{2}} = 56 A C (b) = \sqrt{{(0 + 36)}^{2} + {((0 - 15))}^{2}} = 39$

Thus centroid,

$= (\frac{x_{1} + x_{2} + x_{3}}{3}, \frac{y_{1} + y_{2} + y_{3}}{3}) = (\frac{0 + 20 - 36}{3}, \frac{0 + 15 + 15}{3}) = (- \frac{16}{3}, 10)$

co-ordinates of incentre are-

$= (\frac{a x_{1} + b x_{2} + c x_{3}}{a + b + c}, \frac{a y_{1} + b y_{2} + c y_{3}}{a + b + c}) = (\frac{56 (0) + 20 (39) - 25 (36)}{120}, \frac{56 (0) + 15 (39) + 25 (15)}{120}) = (- 1, 8)$

find the coordinates of the incentre and centroid of the triangle whose sides have the equation 3x -4y=0, 12y+5x=0, y-15=0. also please tell me what is incentre, circumcentre, orthocentre and centroid in a triangle

find the coordinates of the incentre and centroid of the triangle whose sides have the equation 3x -4y=0, 12y+5x=0, y-15=0.

also please tell me what is incentre, circumcentre, orthocentre and centroid in a triangle