Question

An equilateral triangle is drawn by joining the midpoints of the sides of another equilateral triangle.a third triangle is drawn inside the second one joining the midpoints of sides of the second triangle and the process continues infinitely .find the sum of the perimeter of all the equilateral triangle

Answer 1

Answer:

$\red { sum \: of \: the \: perimeter \: of }$

$\red { infinite \: equilateral \: triangles }$

$\green { 6\times side \: of \:the \: first \:triangle}$

Step-by-step explanation:

$Let \: side \: of \: the \: first \: equilateral \\triangle = a$

$Perimeter \: of \: the \: triangle (P_{1}) = 3a$

/*An equilateral triangle is drawn by joining the midpoints of the sides of another equilateral triangle. */

$side \: of \: the \: second \: equilateral \\triangle = \frac{a}{2}$

$Perimeter \: of \: the \: triangle (P_{2}) = \frac{3a}{2}$

$side \: of \: the \: Third \: equilateral \\triangle = \frac{a}{4}$

$Perimeter \: of \: the \: triangle (P_{3}) = \frac{3a}{4}$

$Sum \:of \: infinite \: such \:triangles \\= 3a + \frac{3a}{2} + \frac{3a}{4} + \cdot\cdot\cdot,$

$S_{\infinite } = (3a)[1+\frac{1}{2}+\frac{1}{4}+\cdot\cdot\cdot,$

___________________________

$S = 1+\frac{1}{2}+\frac{1}{4}+\cdot\cdot\cdot,\\is\: infinite \:g.p$

$= \frac{1}{1-\frac{1}{2}}\\=\frac{1}{\frac{1}{2}}\\=2$

$\boxed {\pink { Sum \:of \: infinite \:g.p = \frac{a}{1-r}}}$

$where , \: first \:term = a ,\: common \:ratio (r) =\frac{a_{2}}{a_{1}},\: |r| <1$

__________________________

$= 3a \times 2\\= 6a$

Therefore.,

$\red { sum \: of \: the \: perimeter \: of }$

$\red { infinite \: equilateral \: triangles }$

$\green { 6a}$

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