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Question

Find the coordinates of the incentre and centroid of the triangle whose sides have the equations 3x − 4y = 0, 12y + 5x = 0 and y − 15 = 0.

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Solution

The given lines are as follows:

3x − 4y = 0 ... (1)

12y + 5x = 0 ... (2)

y − 15 = 0 ... (3)

In triangle ABC, let equations (1), (2) and (3) represent the sides AB, BC and CA, respectively.



Solving (1) and (2):
x = 0, y = 0

Thus, AB and BC intersect at B (0, 0).

Solving (1) and (3):
x = 20 , y = 15

Thus, AB and CA intersect at A (20, 15).

Solving (2) and (3):
x = −36 , y = 15

Thus, BC and CA intersect at C (−36, 15).

Let us find the lengths of sides AB, BC and CA.

AB=20-02+15-02=25BC=0+362+0-152=39AC=20+362+15-152=56

Here, a = BC = 39, b = CA = 56 and c = AB = 25
Also, x1, y1 = A (20, 15), x2, y2 = B (0, 0) and x3, y3 = C (−36, 15)

Centroid=x1+x2+x33, y1+y2+y33 =20+0-363, 15+0+153=-163, 10

And, incentre=ax1+bx2+cx3a+b+c, ay1+by2+cy3a+b+c =39×20+56×0-25×3639+56+25, 39×15+56×0+25×1539+56+25 =-120120, 120×8120=-1, 8

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