Question

In a triangle ABC,AD is the median and A(1,2) and B(3,4) then find the centroid of triangle ABC

Answer 1

Centroid of the triangle is (2,3) for the triangle ABC where AD is the median A(1,2), B(3,4) and C (2,3)

Step-by-step explanation:

Given Data

In a triangle ABC,

A = (1,2) and B = (3,4)

To find - C and Centroid of the triangle

In a triangle, AB = BC = AC

Also AB² = BC² = AC²

AB² = (3-1)² + (4-2)²

AB² = 2² + 2² = 4+4

AB² = 8                  --------> (1)

AC² = (x-1)² + (y-2)² = x² +1 - 2 x + y²+ 4- 4 y    -----------> (2)

BC² = (x-3)² + (y-4)² = x²+9 -6x+ y²+ 16 -8y

BC² =  x² -6x+ y² -8y + 25   -----------> (3)

Equate (2) and (3)

AC² = BC²

x² +1 - 2 x + y²+ 4- 4 y = x² -6x+ y² -8y + 25

25 - 6x - 8y = -2x - 4x + 5

25 - 4x =4y +5

4x  + 4y = 20

x + y = 5 -----------> (4)

y = 5 - x ------------> (5)

Equate (1) and (2)

AB² =  AC²

8  = x² +1 - 2 x + y²+ 4- 4 y

Substitute the value y = 5-x in above equation

8 =  x² +1 - 2 x + 25x + x² -10x -20 + 4x

8 = 2x² - 8x + 6

2x² - 8x =2

x² - 4x = 1

x² - 4x - 1= 0

solve the above quadratic equation

x² - 4x - 1 => x² - 2x -2x- 1 = 0

x² - 4x - 1 => (x² - 2x) -(2x + 1) = 0

=> x² - 2x = 0

x² = 2x

x = 2

Substitute the x value in y = 5-x

y = 5 - 2

y = 3

Therefore the point C = ( 2,3)

Centroid of the triangle = $\frac{x_{1}+x_{2}+x_{3}}{3}, \frac{y_{1}+y_{2}+y_{3}}{3}$

where $x_1, y_1$ = (1,2)

$x_2,y_2$ = (3,4)

$x_3,y_3$ = (2,3)

Substitute the respective values in Centroid of the triangle formula

Centroid of the triangle =  $\frac{1+3+2}{3}, \frac{2+4+3}{3}$

Centroid of the triangle = $\frac{6}{3}, \frac{9}{3}$

Centroid of the triangle = 2 , 3

Centroid of the triangle is (2,3) for the triangle ABC where AD is the median A(1,2), B(3,4) and C (2,3)

To learn more ...

1. https://brainly.in/question/5201356

2. https://brainly.in/question/2479570