Question

Question 4 Equilateral triange AB.Co is given. For every n, natural number, An, Bn, Cn are midpoints of sides Bn iCn 1, Cn 1 An 1, An 1Bn 1, respectively. Part of triangle A, B.Co that is outside of triangle A B C shade in red color. On the same way shade part of triangle A2B2C2 outside of triangle A3B3C3 and so on. Continue the process and find percentage of triangle A, B.Co that is shaded.​

Answer 1

Given :  equilateral triangle A₀, B₀, C₀ is given

for every n,  natural numbers  Aₙ , Bₙ , Cₙ   are mid points of  Bₙ₊₁ Cₙ₋₁ , Cₙ₋₁ Aₙ₋₁ and Aₙ₋₁ Bₙ₋₁

Part of triangle A₀B₀C₀ that is outside of triangle A₁B₁C₁ shade in red color

on the same way  A₂B₂C₂ outside of triangle A₃B₃C₃ and so on.

To Find : percentage of triangle A₀B₀C₀ that is shaded. ​

Solution:

Area of  ΔA₀B₀C₀  = A

Then Area of ΔA₁B₁C₁  = A/4  as sides are half

Hence Part of triangle A₀B₀C₀ that is outside of triangle A₁B₁C₁ shade in red color = A - A/4  = 3A/4

Area of  ΔA₂B₂C₂  = (A/4)/4  = A/16

Then Area of Δ A₃B₃C₃ =  (A/16)/4 = A/64    

Hence Part of triangle ΔA₂B₂C₂ that is outside of triangle Δ A₃B₃C₃ shade in red color = A/16 - A/64  = 3A/64

3A/4  then 3A/64  and so on

This is an GP

with a  = 3A/4

r = 1/16

Sₙ = a/(1 - r)  for infinite series   r < 1

Hence  Sum   =  (3A/4)/( 1 - 1/16)    

= 12A/15

= 4A/5

= 80 %

80 % of     triangle A₀B₀C₀   is shaded.

Learn More:

In an infinite g.P. Each term is equal to three times the sum of all the ...

brainly.in/question/9079152

if s1,s2,s3...sp are the sum of infinite geometric series whose first ...

brainly.in/question/5796750