Question

please give brief solution ​

Answer 1

Required Answer:-

Since D, E and F are the midpoints of sides AB, AC and BC, by converse basic proportionality theoram,

• DE || BC & DE = BC / 2
• DF || AC & DF = AC / 2
• EF || AB & EF = AB / 2

Then,

⇒ Perimeter of ∆ ABC = AB + BC + CA/2

⇒ And, perimeter of ∆DEF = DE + EF + FD

From the relation we got,

⇒ Perimeter of ∆DEF = AB/2 + BC/2 + CA/2

⇒ Perimeter of ∆DEF = AB + BC + CA / 2

That means, ratio of perimeters of ∆ABC and ∆DEF will be 2 : 1. (B)

What was your mistake?

• You might be confused since there are 4 triangles inside the bigger triangle looking similar in size.
• The area ratio will be 4 : 1. That is what you mistaken from the diagram$.$

Answer 2

Required solution :

★ We are able to see in the above attachment that there is a big triangle drawn and inside it there is a small triangle is drawn too.

★ And here point D, E and F are the mid-point'(s) of the side AB, AC and BC, respectively of an equilateral triangle i.e., equilateral triangle is that type of triangle whose all the three sides are equal. ∆ABC, the ratio of the perimeter of ∆ABC to that of ∆DEF is

• a). 3:1
• b). 2:1
• c). 1:1
• d). 4:1

Henceforth,

DE||BC and DE = BC/2

DF||AC and DF = AC/2

EF||AB and EF = AB/2

Now,

»»» Perimeter of ∆ABC = AB + BC + CA/2

»»» Perimeter of ∆DEF = DE + EF + FD

According to the above formula,

»»» Perimeter of ∆DEF = AB/2 + BC/2 + CA/2

»»» Perimeter of ∆DEF = AB + BC + CA / 2

• ∆ABC, the ratio of the perimeter of ∆ABC to that of ∆DEF is 2:1, Option b).

• That means, ratio of perimeters of ∆ABC and ∆DEF will be 2:1, Option b).