Question

in fig d is the midpoint of bc and e is the midpoint of ad then prove that area of triangle abc / area of triangle bed =4​

Answer 1

Answer:

Proved.

Area ΔBED=  $\dfrac{1}{4}$ ΔABC.

AccuraTe QuesTion :

ABC is a triangle in which D is the midpoint of BC and E is the midpoint of AD. Prove that : Area ΔBED = $\dfrac{1}{4}$ ΔABC.

SoluTion :

Given :

• ABC is a triangle.
• D is the midpoint of BC and E is the midpoint of AD.

Proof :

AD is the median of ΔABC. Therefore, it will divide ΔABC into two triangles of equal area.

Therefore,

Area (ΔABD) = Area (ΔACD)

Now,

⇒ Area (ΔABD ) = $\dfrac{1}{4}$ area (Δ ABC)

_______________ [ Equation - 1 ]

In ΔABD, E is the mid-point of AD.

∴ BE is the median.

$\sf\\\implies Area\;\triangle BED = Area\;\triangle ABE\\ \\ \implies Area\;\triangle BED = \dfrac{1}{2}\;Area\;\triangle ABE\\ \\ \implies Area\; \triangle BED = \dfrac{1}{2} \times \dfrac{1}{2} Area\;\triangle ABC\\ \\ \therefore Area\; \triangle BED = \dfrac{1}{4}\;Area\;\triangle ABC$

Proved.

Answer 2

Answer:

In ABC ,

AD is median

So ,AD divide ABC into 2 triangles of equal area

that is, Area of ABD =Area of ADC

Also, BE is median of ABD

So, It wil divide ABD into 2 triangles of equal area

that is, Area of ABE = Area of BED

Also, Area of ABD = Area of ABE + Area of BED

As Area of ABE = Area of BED

So, Area of ABD = Area of BED + Area of BED

Area of ABD = 2 (Area of BED)

As , Area of ABD =Area of ADC

And Area of ABC = Area of ABD + Area of ADC

So, Area of ABC = 2 Area of ABD

As Area of ABD = 2(Area of BED)

So, Area of ABC = 2 (2(Area of BED))

Area of ABC = 4 (Area of BED)

Area of ABC/Area of BED = 4

Hence proved