# In a triangle ABC, E is the midpoint of median AD. Show that $area({\text{BED) =

}}\frac{1}{4}area({\text{ABC)}}{\text{.}}$

Last updated date: 23rd Mar 2023

•

Total views: 307.8k

•

Views today: 3.87k

Answer

Verified

307.8k+ views

Hint: - Here we go through the properties of the median of the triangle to prove the question.As we know that the median of the triangle bisects the triangle into two equal parts.

Here in the question it is given that,

AD is the median on BC of $\vartriangle {\text{ABC}}$ and we know that the median divides the triangle in two equal parts.

$\therefore area(\vartriangle {\text{ABD) = }}area(\vartriangle {\text{ACD)}}{\text{.}}$

It is also written as,

$\therefore area(\vartriangle {\text{ABD) = }}area(\vartriangle {\text{ACD) =

}}\frac{1}{2}area(\vartriangle {\text{ABC)}}$……… (1)

Now in the question it is given that EB is the median on AD of$\vartriangle {\text{ABD}}$.

$\therefore area(\vartriangle {\text{BED) = }}area(\vartriangle {\text{BEA)}}$ As we know the property of the median above.

And it is also written as,

$\therefore area(\vartriangle {\text{BED) = }}area(\vartriangle {\text{BEA) =

}}\frac{1}{2}area(\vartriangle {\text{ABD)}}$………… (2)

By the equation (1) and (2) we can write,

$\therefore area(\vartriangle {\text{BED) = }}\frac{1}{2} \times \frac{1}{2}area(\vartriangle

{\text{ABC)}}$ $\because area(\vartriangle {\text{ABD) = }}\frac{1}{2}area(\vartriangle {\text{ABC)}}$

As we proved above.

$\therefore area(\vartriangle {\text{BED) = }}\frac{1}{4}area(\vartriangle {\text{ABC)}}$ Hence, proved.

Note: - Whenever we face such a type of question the key concept of solving the question is to first make the diagram and name it as given in the question. Then apply the property of that statement which is given in the question here in this question we apply the property of median.

Here in the question it is given that,

AD is the median on BC of $\vartriangle {\text{ABC}}$ and we know that the median divides the triangle in two equal parts.

$\therefore area(\vartriangle {\text{ABD) = }}area(\vartriangle {\text{ACD)}}{\text{.}}$

It is also written as,

$\therefore area(\vartriangle {\text{ABD) = }}area(\vartriangle {\text{ACD) =

}}\frac{1}{2}area(\vartriangle {\text{ABC)}}$……… (1)

Now in the question it is given that EB is the median on AD of$\vartriangle {\text{ABD}}$.

$\therefore area(\vartriangle {\text{BED) = }}area(\vartriangle {\text{BEA)}}$ As we know the property of the median above.

And it is also written as,

$\therefore area(\vartriangle {\text{BED) = }}area(\vartriangle {\text{BEA) =

}}\frac{1}{2}area(\vartriangle {\text{ABD)}}$………… (2)

By the equation (1) and (2) we can write,

$\therefore area(\vartriangle {\text{BED) = }}\frac{1}{2} \times \frac{1}{2}area(\vartriangle

{\text{ABC)}}$ $\because area(\vartriangle {\text{ABD) = }}\frac{1}{2}area(\vartriangle {\text{ABC)}}$

As we proved above.

$\therefore area(\vartriangle {\text{BED) = }}\frac{1}{4}area(\vartriangle {\text{ABC)}}$ Hence, proved.

Note: - Whenever we face such a type of question the key concept of solving the question is to first make the diagram and name it as given in the question. Then apply the property of that statement which is given in the question here in this question we apply the property of median.

Recently Updated Pages

Calculate the entropy change involved in the conversion class 11 chemistry JEE_Main

The law formulated by Dr Nernst is A First law of thermodynamics class 11 chemistry JEE_Main

For the reaction at rm0rm0rmC and normal pressure A class 11 chemistry JEE_Main

An engine operating between rm15rm0rm0rmCand rm2rm5rm0rmC class 11 chemistry JEE_Main

For the reaction rm2Clg to rmCrmlrm2rmg the signs of class 11 chemistry JEE_Main

The enthalpy change for the transition of liquid water class 11 chemistry JEE_Main

Trending doubts

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Write a letter to the Principal of your school to plead class 10 english CBSE