Ex 6.4, 5
D, E and F are respectively the mid-points of sides AB, BC and CA of ΞABC. Find the ratio of the areas of ΞDEF and ΞABC.
Given: Ξ ABC
& D,E,F mid-points of AB,BC & CA respectively
To find: (ππ βπ·πΈπΉ)/(ππ βπ΄π΅πΆ)
Note: Since we need to find ratio of area of ΞDEF and ΞABC.
We first need to prove these triangles are similar
Solution:
We know that
line joining mid-points of two sides of a triangle
is parallel to the 3rd side
In ΞABC ,
D and F are mid-points of AB and AC resp.,
β΄ DF β₯ BC
So, DF β₯ BE also
Similarly,
E and F are mid-points of BC and AC resp.
EF β₯ AB
Hence, EF β₯ DB
From (1) & (2)
DF β₯ BE & FE β₯ DB
Therefore, opposite sides of quadrilateral is parallel
DBEF is a parallelogram
DBEF is a parallelogram
Now we know that ,
in parallelogram, opposite angle are equal
Hence β DFE =β ABC
Similarity,
we can prove DECF is a parallelogram
In a parallelogram, opposite angles are equal
Hence, β EDF= β ACB
Now , in ΞEDF and ΞABC
β DFE =β ABC
β EDF= β ACB
By using AA similarity criterion
Ξ DEF βΌ Ξ ABC
We know that if two triangles are similar,
the ratio of their area is always equal to
the square of the ratio of their corresponding side
β΄ (ππ βπ·πΈπΉ)/(ππ βπ΄π΅πΆ) = π·πΈ2/π΄πΆ2
(ππ βπ·πΈπΉ)/(ππ βπ΄π΅πΆ) = πΉπΆ2/π΄πΆ2
(π΄πππ ππ βπ·πΈπΉ)/(π΄πππ ππ βπ΄π΅πΆ)=( π΄πΆ/2 )^2/(π΄πΆ)2
(π΄πππ ππ βπ·πΈπΉ)/(π΄πππ ππ βπ΄π΅πΆ)=((π΄πΆ)2/4)/(π΄πΆ)2
(π΄πππ ππ βπ·πΈπΉ)/(π΄πππ ππ βπ΄π΅πΆ)=(1/4)/1
Hence , (π΄πππ ππ β π·πΈπΉ)/(π΄πππ ππ β π΄π΅πΆ)=1/4

Made by

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.