Question

vertices of a triangle are (3,1), (-2,3) (-1,-2) find centroid ilof triangle and area of triangle ​

Answer 1

Given: A triangle formed by the points (3, 1), (-2, 3) and (-1, -2).

To find: The centroid and the area of the triangle.

Answer:

Let's first find the centroid.

Formula to find the centroid of the triangle:

$\tt Centroid\ =\ \Bigg(\dfrac{x_1\ +\ x_2\ +\ x_3}{3},\ \dfrac{y_1\ +\ y_2\ +\ y_3}{3}\Bigg)$

From the given points, we have:

$\tt x_1\ =\ 3\\\\x_2\ =\ -2\\\\x_3\ =\ -1\\\\y_1\ =\ 1\\\\y_2\ =\ 3\\\\y_3\ =\ -2$

Using them in the formula,

$\tt Centroid\ =\ \Bigg(\dfrac{3\ +\ -2\ +\ -1}{3},\ \dfrac{1\ +\ 3\ +\ -2}{3}\Bigg)\\\\\\Centroid\ =\ \Bigg(\dfrac{3\ -\ 3}{3},\ \dfrac{2}{3}\Bigg)\\\\\\\bf Centroid\ =\ \Bigg(0,\ \dfrac{2}{3}\Bigg)$

Now, let's find the area of the triangle.

Formula to find the area of a triangle:

$\tt Area\ =\ \dfrac{1}{2}\ \times\ \Bigg(x_1\Big(y_2\ -\ y_3\Big)\ +\ x_2\Big(y_3\ +\ y_1\Big)\ +\ x_3\Big(y_1\ -\ y_2\Big)\Bigg)$

Again, as earlier,

$\tt x_1\ =\ 3\\\\x_2\ =\ -2\\\\x_3\ =\ -1\\\\y_1\ =\ 1\\\\y_2\ =\ 3\\\\y_3\ =\ -2$

Using them in the formula,

$\tt Area\ =\ \dfrac{1}{2}\ \times\ \Bigg(3\Big(3\ +\ 2\Big)\ +\ -2\Big(-2\ -\ 1\Big)\ +\ -1\Big(1\ +\ 3\Big)\Bigg)$

$\tt Area\ =\ \dfrac{1}{2}\ \times\ \Bigg(3\Big(5\Big)\ +\ -2\Big(-3\Big)\ +\ -1\Big(4\Big)\Bigg)\\\\\\Area\ =\ \dfrac{1}{2}\ \times\ \Bigg(15\ +\ 6\ -\ 4\Bigg)\\\\\\Area\ =\ \dfrac{1}{2}\ \times\ 17\\\\\\Area\ =\ \dfrac{17}{2}\\\\\\\bf Area\ =\ 8.5\ units$

Therefore, the centroid of a triangle formed by the points (3, 1), (-2, 3) and (-1, -2) is (0, 2/3) and its area is 8.5 units.

Answer 2

Answer:

Here goes the solution!

Finding centroid,

Centroid = [ x1 + x2 + x3/3 ; y1 + y2 + y3/3 ]

So, x1 = 3 ; x2 = -2 ; x3 = -1

y1 = 1 ; y2 = 3 ; y3 = -2

Using formula,

Centroid = [ 3 + (-2) + (-1)/3 ; 1 + 3 + (-2)/3 ]

= [ 3 - 3/3 ; 2/3 ]

= [ 0 ; 2/3 ]

Finding area of triangle,

Area of triangle = 1/2[x1 (y2 - y3) + x2 (y3 + y1) + x3 (y1 - y2)]

Again, x1 = 3 ; x2 = -2 ; x3 = -1

y1 = 1 ; y2 = 3 ; y3 = -2

Substituting in formula,

Area = 1/2[3 (3 - (-2) + (-2)( (-3) + 1) + (-1)(1 - 3) ]

= 1/2[3 (5) + (-2)(-3) + (-1)(4) ]

= 1/2[ 15 + 6 - 4 ]

= 1/2 × 17

= 17/2

= 8.5 units

Hence, the centroid of the triangle is (0 , 2/3) & area of the triangle is 8.5 units.