
In the given figure, $E$ is any point on median $AD$ of a $\Delta ABC$. Show that $area\;\left( {ABE} \right) = area\;\left( {ACE} \right)$.
Answer
509.1k+ views
Hint:In the solution we will use the concept of median which states that median divides a triangle into two triangles of equal area. The statistical median basically used when the large numbers of data are provided.
Complete step-by-step solution
Given: $ABC$ is a triangle with $AD$ as median that is $BD = CD$ and $E$ is any point on $AD$.
Since $AD$ is a median of $\Delta ABC$. Thus, it will divide $\Delta ABC$ into two triangles of equal area.
$area\;\left( {ABD} \right) = area\;\left( {ACD} \right)$……(1)
Also, $ED$ is a median of $\Delta ABC$. Thus, it will divide $\Delta ABC$ into two triangles of equal area.
$area\;\left( {EBD} \right) = area\;\left( {ECD} \right)$……(2)
Subtracting equation (2) from (1) we get
$\begin{array}{c}
area\;\left( {ABD} \right) - area\left( {EBD} \right) = area\;\left( {ACD} \right) - area\left( {ECD} \right)\\
area\;\left( {ABE} \right) = area\;\left( {ACE} \right)
\end{array}$
Therefore, $area\;\left( {ABE} \right) = area\;\left( {ACE} \right)$ is proved.
Note: Make sure to use the median of the triangle instead of statistical median. The median of the triangle tells us about the area of two triangles which are bisected by median. In a triangle, a median is the line which connects the vertex of a triangle with the midpoint of the line which is opposite to the vertex. The median in statics is used if the large numbers of data are given.
Complete step-by-step solution
Given: $ABC$ is a triangle with $AD$ as median that is $BD = CD$ and $E$ is any point on $AD$.

Since $AD$ is a median of $\Delta ABC$. Thus, it will divide $\Delta ABC$ into two triangles of equal area.
$area\;\left( {ABD} \right) = area\;\left( {ACD} \right)$……(1)
Also, $ED$ is a median of $\Delta ABC$. Thus, it will divide $\Delta ABC$ into two triangles of equal area.
$area\;\left( {EBD} \right) = area\;\left( {ECD} \right)$……(2)
Subtracting equation (2) from (1) we get
$\begin{array}{c}
area\;\left( {ABD} \right) - area\left( {EBD} \right) = area\;\left( {ACD} \right) - area\left( {ECD} \right)\\
area\;\left( {ABE} \right) = area\;\left( {ACE} \right)
\end{array}$
Therefore, $area\;\left( {ABE} \right) = area\;\left( {ACE} \right)$ is proved.
Note: Make sure to use the median of the triangle instead of statistical median. The median of the triangle tells us about the area of two triangles which are bisected by median. In a triangle, a median is the line which connects the vertex of a triangle with the midpoint of the line which is opposite to the vertex. The median in statics is used if the large numbers of data are given.
Recently Updated Pages
Master Class 9 General Knowledge: Engaging Questions & Answers for Success

Master Class 9 English: Engaging Questions & Answers for Success

Master Class 9 Science: Engaging Questions & Answers for Success

Master Class 9 Social Science: Engaging Questions & Answers for Success

Master Class 9 Maths: Engaging Questions & Answers for Success

Class 9 Question and Answer - Your Ultimate Solutions Guide

Trending doubts
Difference Between Plant Cell and Animal Cell

What is the Full Form of ISI and RAW

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Discuss what these phrases mean to you A a yellow wood class 9 english CBSE

Name 10 Living and Non living things class 9 biology CBSE

Name the states which share their boundary with Indias class 9 social science CBSE
