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One side of an equliateral triangle is 3...

One side of an equliateral triangle is 32 cm . The mid -points of its sides are joined to form
another trianbgle whose mid -point are in turn joinded to form still another triangle . This process
continus indefinitely . Then find the sum of the perimeters and areas of all the triangles .

Text Solution

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Let a be the side of ` Delta ABC` , then a = 32 cm

Then sies of ` Delta A_(1) B_(1) C_(1)` formed by joining the mid-points of the sides of ` Delta ABC is (a)/(2)` and so on
` therefore ` Sum of perimeters of all triangles
` = 3a + (3a)/(2) + (3a)/(4) + (3a)/(8) + ... ` upto ` oo` terms
` = (3a)/(1-(1)/(2))= 6a = 6 xx32 = 192 ` cm
Sum of areas of all such triangles
`(sqrt(3))/(4) a^(2) + (sqrt(3))/(4) ((a)/(2))^(2) + (sqrt(3))/(4) ((a)/(4))^(2) + (sqrt(3))/(4) ((a)/(8))^(2) + ..."to" oo`
`(sqrt(3))/(4) [(a^(2))/(1-(1)/(4))]= (sqrt(3))/(4) (a^(2) xx 4)/(3) = sqrt(3)x (32 xx32)/(3)`
` (1024)/(3)sqrt(3)` . square units .
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