Question

In the given figure; AD is the median of \[\Delta {\text{ABC}}\] and E is any point on median AD. Prove that \[{\text{Area}}\left( {\Delta {\text{ABE}}} \right) = {\text{Area}}\left( {\Delta {\text{ACE}}} \right)\].

Answer 1

Hint:
First, we will consider triangle \[\Delta {\text{ABC}}\], as D is the midpoint of BC, so we will have that AD is median so the median divides a triangle into two equal areas. We will now consider triangle \[\Delta {\text{EBC}}\], as D is the midpoint of BC, so we will have that ED is median, which divides a triangle into two equal areas. Then we will find the difference to find the required value.

Complete step by step solution:
We are given that AD is the median of \[\Delta {\text{ABC}}\] and E is any point on median AD.
In triangle \[\Delta {\text{ABC}}\],
 We have D is the midpoint of BC, so we will have that AD is median.
Since we know that the median divides a triangle into two equal areas, so we get
\[{\text{Area}}\left( {\Delta {\text{ABD}}} \right) = {\text{Area}}\left( {\Delta {\text{ABC}}} \right){\text{ ......eq.(1)}}\]
In triangle \[\Delta {\text{EBC}}\],
We have D is the midpoint of BC, so we will have that ED is median.
Since we know that the median divides a triangle into two equal areas, so we get
\[{\text{Area}}\left( {\Delta {\text{EBD}}} \right) = {\text{Area}}\left( {\Delta {\text{EDC}}} \right){\text{ ......eq.(2)}}\]
Subtracting the equation (1) from equation (2), we get
\[
   \Rightarrow {\text{Area}}\left( {\Delta {\text{ABD}}} \right) - {\text{Area}}\left( {\Delta {\text{EBD}}} \right) = {\text{Area}}\left( {\Delta {\text{ADC}}} \right) - {\text{Area}}\left( {\Delta {\text{EDC}}} \right) \\
   \Rightarrow {\text{Area}}\left( {\Delta {\text{ABE}}} \right) = {\text{Area}}\left( {\Delta {\text{ACE}}} \right) \\
 \]

Hence, proved.

Note:
In solving these types of questions, first, draw the pictorial representation of the given problem for better understanding. You need to know the properties of triangles and their midpoint. Then we will use the properties accordingly. This is a really simple problem, one should only need to know the definitions and crack the point of using the ASA congruence rule. It is clear from the diagram that it is a triangle.