Find the point that makes two triangle equal

Question

Can anyone help me with this problem? I draw the graph according to the problem, but don't know how to solve the problem. Thanks.

A triangle has vertices with coordinates A(0,15), B(0,0), and C(10,0). Find the coordinates of point D on AC so that the area of triangle ABD is equal to the area of triangle DBC.

Answer 1

Hint: Let $BX$ be perpendicular to $AC$, $ABD$ and $DBC$ have the same altitude and since $[ABD]=[DBC]$ we get $AD=DC$. Use mid-point formula.

Answer 2

$(x,y )$ is a coordinat which divided triangle into two equal area. $$\Delta= \frac12\begin{vmatrix} 0 & 15 \\ 0& 0 \\ x& y\\ 0 &15 \end{vmatrix}= \frac12\begin{vmatrix} 0 & 0 \\ 10& 0 \\ x& y\\ 0 &0 \end{vmatrix}$$ $$ 15x= 3y =\text{Half of area of triangle } \Delta ABC$$

Answer 3

When $BD$ is the median then the area of $\Delta ABD$ is equal to the area of $\Delta BDC$.
Therefore the coordinates of $D$ is $(5, 7.5)$.