Question

$\sf{s_1 , s_2} \: are \: inscribed \: and \: circumscribed \: circles \: of \: a \: Triangle \: with \: sides 3,4,5 \: then \sf\dfrac{area\:of\:s_1}{area\:of\:s_2} \ \textless \ br /\ \textgreater$
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Answer 1

$\Huge\text{ \dfrac{S_{1}}{S_{2}}=\dfrac{4}{25} }$

$\huge\textbf{Question}$

$\textrm{A triangle has side lengths 3, 4, 5.}$

$\textrm{ S_{1} is the area of the inscribed circle.}$

$\textrm{ S_{2} is the area of the circumscribed circle.}$

$\textrm{Find \dfrac{S_{1}}{S_{2}} .}$

$\huge\textbf{Explanation}$

$\textrm{We know that (3,4,5) is a Pythagorean triple.}$

$\textrm{ \therefore Given triangle is a right triangle.}$

$\Large\textrm{Inradius and the Area}$

$\cdots\longrightarrow S=\dfrac{1}{2}\times r\times(a+b+c)$

$\textrm{We know that the area of the triangle is 6.}$

$\large\textrm{(By formula)}$

$6=\dfrac{1}{2}\times r\times(3+4+5)$

$6=\dfrac{1}{2}\times r\times12$

$6=6r$

$r=1$

$\Large\textrm{Properties of Circumcircle}$

$\textrm{For a right triangle, -}$

$\textrm{it is located at the midpoint of the hypotenuse.}$

$\textrm{Hence, the radius is half the hypotenuse, \dfrac{5}{2} .}$

$\textrm{Lastly, -}$

$\dfrac{S_{1}}{S_{2}}=\dfrac{1^{2}}{(5/2)^{2}}$

$\dfrac{S_{1}}{S_{2}}=\dfrac{4}{25}$

$\huge\textbf{Answer}$

$\textrm{Find \dfrac{S_{1}}{S_{2}} .}$

$\textrm{The required answer is \dfrac{S_{1}}{S_{2}}=\dfrac{4}{25} .}$

Answer 2

QUESTION :-

A triangle has side lengths 3, 4, 5.S₁ is the area of the inscribed circle. S₂ is the area of the circumscribed circle.Find S1 /S2

GIVEN :-

A triangle has side lengths 3, 4, 5

S₁ is the area of the inscribed circle

S₂ is the area of the circumscribed circle

TO FIND :-

Find S1 /S2 = ?

SOLUTION :-

first we have formula :-

r = area of traingle / semi - perimeter of traingle

R = Length of hypotenuse / 2 × sine of central angle

ACCORDING TO THE QUESTION :-

= 1/2 × 3 × 4 / 3 + 4 + 5 / 2 6/6 = 1

then, we have :-

R = Length of hypotenuse / 2 × sine of central angle

= 5 / 2 × sin 90° = 5/2

ratio of area is

πr² / πR² = 1² / ( 5/2) = 1/25 /4 = 4/25

so, S1/S2 = 4/25