Question

One side of an equilateral triangle is 24 cm. The midpoints of its sides are joined to form another triangle whose midpoints are in turn joined to form still another triangle. This process continues indefinitely. Find the sum of the perimeters of all the triangles.​

Answer 1

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• The mid-points of its sides are joined to form another triangle whose mid-points are joined to form another triangle. ... =144 which is the sum of perimeter of all triangle.

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Answer 2

$\large\underline{\sf{Solution-}}$

It is given that,

• Side of an equilateral triangle is 24 cm

We know,

• Perimeter of equilateral triangle = 3 × side

$\red{\bf :\longmapsto\:P_1 = 24 \times 3 = 72 \: cm - - - (1)}$

Now,

Another equilateral triangle is formed by joining midpoints.

So,

• Side of this equilateral triangle = 12 cm

Since,

• Perimeter of equilateral triangle = 3 × side

$\red{\bf :\longmapsto\:P_2 = 12 \times 3 = 36 \: cm - - - (2)}$

Again,

Another equilateral triangle is formed by joining midpoints.

So,

• Side of this equilateral triangle = 6 cm

Since,

• Perimeter of equilateral triangle = 3 × side

$\red{\bf :\longmapsto\:P_3 = 6 \times 3 = 18 \: cm - - - (3)}$

Hence,

The sum of perimeters of all equilateral triangle thus formed is

$\red{\rm :\longmapsto\: \bf{ \: P_1 + P_2 + P_3 + - - - - }}$

$\rm \: \: = \: \: 72 + 36 + 18 + - - - - -$

Its forms a GP series,

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ Sum of infinite terms of an arithmetic sequence is,

$\boxed{\red{\bigg \{\bf \: S_ \infty = \dfrac{a}{1 - r} \: \: where \: |r| < 1 \bigg \}}}$

Wʜᴇʀᴇ,

• a is the first term of the sequence.

• r is the common ratio

Tʜᴜs,

From the given series,

$\rm :\longmapsto\:a = 72$

$\rm :\longmapsto\:r = \dfrac{1}{2}$

So,

$\rm :\longmapsto\:S_ \infty = \dfrac{72}{1 - \dfrac{1}{2} }$

$\rm :\longmapsto\:S_ \infty = \dfrac{72}{ \dfrac{2 - 1}{2} }$

$\rm :\longmapsto\:S_ \infty = \dfrac{72}{ \dfrac{1}{2} }$

$\bf\implies \:S_ \infty = 72 \times 2 = 144$

Hence,

• Sum of perimeters of all equilateral triangles thus formed = 144 cm

Additional Information :-

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ nᵗʰ term of an geometric sequence is,

$\boxed{ \bf \: a_n \: = \: {ar}^{n - 1}}$

Wʜᴇʀᴇ,

• aₙ is the nᵗʰ term.

• a is the first term of the sequence.

• n is the no. of terms.

• r is the common ratio.

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ Sum of first n terms of an grometric sequence is,

$\boxed{ \bf \: S_n = \dfrac{a( {r}^{n} - 1)}{r - 1} \: \: provided \: that \: r \ne \: 1}$

Wʜᴇʀᴇ,

• Sₙ is the sum of n terms.

• a is the first term of the sequence.

• n is the no. of terms.

• r is the common ratio.