Question

d , e , f are respectively the mid points of sides ab, bc and ca of triangle abc . find the ratio of the areas of triangle def and triangle abc​

Answer 1

Answer:

Step-by-step explanation:

Given :-

In △ABC, D, E and F are the mid points of sides AB, BC and CA.

To Find :-

ar(△DEF)/ar(△ABC) = ?

Solution :-

From mid-point theorem, we get

DE || AC and DE = $\sf\dfrac{1}{2}$AC ....... (i)    

DF || BC and DF = $\sf\dfrac{1}{2}$BC .........(ii)

EF || AB and EF = $\sf\dfrac{1}{2}$AB ..........(iii)    

From the equation (i), (ii) and (iii), we get

$\sf\implies \dfrac{DE}{AC}=\dfrac{DF}{BC}=\dfrac{EF}{AB}=\dfrac{1}{2}$

⇒ △DEF ∼ △ABC

[By SSS Similarity Criterion]

$\bf\implies Area\Delta(\dfrac{DE}{ABC})=(\dfrac{DE}{AC})^2=(\dfrac{1}{2})^2=\dfrac{1}{4}$

Answer 2

$\huge\mathscr\blue{\underline{Answer\: =\:1:4}}$

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$\sf\green{\underline{Given:}}$

• In ΔABC, D, E and F are are the mind points of side AB, BC and CA.

$\sf\green{\underline{To\:Find:-}}$

• The ratio of the areas of ΔDEF and ΔABC,

$\textbf{\underline{\underline{SolUtion:}}}$

As we know, the line joining the mid points of two sides of a Δ is ║ to the third side and also, half of it.

So,

BC = FC

Also,

$\sf {DE} =\dfrac{1}{2} \times{BC} \\\\ \implies \dfrac{DE}{BC} =\dfrac{1}{2}$ ___________ ( ! )

Similarly for Side DF and BC,

$\sf \dfrac{DF}{AC}=\dfrac{1}{2}$ ___________ ( !! )

Also, For side EF and BC,

$\sf\dfrac{EF}{AB}=\dfrac{1}{2}$ __________ (  !!! )

Now,

From eq. ( ! ), ( !! ) and ( !!! ),

It can be said that,

$\sf\dfrac{DE}{BC}=\dfrac{DF}{AC}=\dfrac{EF}{AB}=\dfrac{1}{2}$

We've a property, according to which if two triangles are given and side of one Δ is proportional to the other triangle's side, then their corresponding angles will be equal. If corresponding angles will be equal then the triangles would be similar, i.e

ΔABC   ~ ΔDEF _________ ( By similarity theorem, ' SSS ' )

So,

⇒ Area of Δ DEF / Area of Δ ABC  $\sf=\dfrac{(DE)^2}{(BC)^2}$

$\sf =\dfrac{1}{2^2}\\\\ =\dfrac{1}{4}$

Hence,

The ratio of the area of the triangle DEF and ABC is 1 : 4

$\rule{200}2$