Question

The three side of a Triangle
AB BC, and
CA are 7, 10, and 12 respectively.
AB is extended to D and AD = 28CM
BC is extended to E and BE = 20 CA
is extended to f and cf = 36 cm
What is the area of triange DEF?​

Answer 1

Step-by-step explanation:

Solution: Recall that the line joining the midpoints of two sides of a triangle is parallel to third side and half of it.

Hence, DF=AC2

⇒DFAC=12−−−−−−(i)

Similarly, EFBC=12−−−−−−(ii)

and DEAB=12−−−−−−(iii)

From equation (i),(ii) and (iii), we have

DFAC=EFBC=DEAB=12

But if in two triangles, sides of one triangle are proportional to the sides of the other triangle, then their corresponding angles are equal and hence the two triangles are similar.

Hence, △ABC∼△DEF(By SSS similarity theorem)

⇒ar(△DEF)ar(△ABC)=EF2BC2

⇒ar(△DEF)ar(△ABC)=(12)2 (By using equation (ii))

⇒ar(△DEF)ar(△ABC)=14

⇒ar(△DEF)=ar(△ABC)4

⇒ar(△DEF)=204

⇒ar(△DEF)=5 cm2