Question

A farmer has a triangular plot whose vertices are A, B, C as shown in figure. He wants to
divide his plot equally to his son and daughter.
Using the concept of median of a triangle, suggest an idea to help the farmer. Then how
much area of land each one gets?

Answer 1

Given:

A triangular plot with vertices A, B and C.

To Find:

Area of two triangles formed by the median are equal.

Solution:

From the graph attached,

A, B and C are the vertices of the given triangle.

Point D is the midpoint of side BC.

AD is the median drawn from point A to side BD of the triangle.

Coordinates of the vertices are,

A(4, 7), B(1, 1) and C(7, 3)

Coordinates of D = $(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$

                             = $(\frac{1+7}{2},\frac{1+3}{2})$

                             = (4, 2)

Area of a triangle with vertices $(x_1,y_1), (x_2,y_2)$ and $(x_3,y_3)$ is given by,

Area = $\frac{1}{2}[x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)]$ (Absolute value)

Area of ΔABD = $\frac{1}{2}[4(1-2)+1(2-7)+4(7-1)]$

                                = $\frac{1}{2}[-4-5+24]$

                                = 7.5 square units

Area of ΔACD = $\frac{1}{2}[4(3-2)+7(2-7)+4(7-3)]$

                        = $\frac{15}{2}$

                        = 7.5 square units

Therefore, area of ΔABD = area of ΔACD