If the centroid and two vertices of the triangle are (4, 8), (9, 7) and (1, 4) respectively, then find the area of the triangle.

Given:

Coordinate of the centroid = (4,8)

Coordinate of the vertex 1 = (9,7)

Coordinate of the vertex 2 = (1,4)

Concept used:

If the coordinates of the vertices of a triangle are (x1, y1), (x2, y2), (x3, y3), then the formula for the centroid of the triangle is given below:

The centroid of a triangle = ((x1 + x2+ x3)/3, (y1+ y2 + y3)/3)

 \(Area = \frac{1}{2}\left| {\begin{array}{*{20}{c}} {{x_1}}&{{y_1}}&1\\ {{x_2}}&{{y_2}}&1\\ {{x_3}}&{{y_3}}&1 \end{array}} \right| \)

Calculation:

Let the coordinate of the third vertex be (a,b).

According to the question,

(a + 9 + 1) ÷ 3 = 4

⇒ a = 2

(b + 7 + 4) ÷ 3 = 8

⇒ b = 13

So, the coordinate of the third vertex is (2,13)
Coordinates of three vertices of triangle are (9,7) , (2,13) & (1,4).
We can calculate the area of a triangle By the Vertex formula.

\(A = \frac{1}{2}\left| {\begin{array}{*{20}{c}} 9&7&1\\ 2&{13}&1\\ 1&4&1 \end{array}} \right|\ \)

So, the area of the triangle

A = (1/2) [9(13 - 4) + 2(4 - 7) + 1(7 - 13)] = 34.5 Unit²

∴ The area of triangle is 34.5 unit².