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If D ,E ,F are midpoints of sides BC ,CA , and AB respectively of ΔABC , then the ratio of the areas of triangles DEF and ABC is
(a) 2 : 3 (b) 1 : 4 (c) 1 : 2 (d) 4 : 5
Answers
Step-by-step explanation:
Given
D,E and F are respectively the midpoints of sides AB,BC and CA of ΔABC
To find
ar(△ABC)
ar(△DEF)
We know that
The line segment joining the midpoints of any two sides of a triangle is half the third side and parallel to it.
∴FD=
2
1
BC,ED=
2
1
ACandEF=
2
1
AB
In △ABC and △EFD, we have
EF
AB
=
FD
BC
=
ED
AC
=2...(i)
⇒△ABC∼△EFD[by SSS similarity criterion]
Also, We know that
If two triangles are similar, then the ratio of the area of both triangles is equal to the square of the ratio of their corresponding sides
∴
ar(△EFD)
ar(△ABC)
=(
EF
AB
)
2
=4
[from (i)]
⇒
ar(△ABC)
ar(△EFD)
=
4
1
Hence, the ratio of the areas of △DEF and △ABC is 1:4
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1:4
Step-by-step explanation:
The ratio of area(ΔDEF): area(ΔABC) is 1 : 4
Among the given options option (a) is 1 :4 is the correct answer. Step-by-step explanation: Given: D, E and F are the mid-points of the sides AB, BC and CA of the ΔABC.