Question

the lenght of a tangent from a point A at a distance 70 cm from the centre of a circle is 56 cm .find the circumference of the circle. ​

Answer 1

$\boxed{ \red{AnswEr:-}}$

• 264 cm

★ GivEn :-

• The length of a tangent from point A at a distance of 70 cm from the centre of a circle is 56 cm

★ To Find :-

• The circumference of the circle

$\boxed{ \red{Solution :-}}$

➠ The length of the tangent, AB = 56 cm

➠ The distance from centre of circle to the tangent, OA = 70 cm

➠ Let the radius of the circle be, OB = r cm

➠ ∆AOB is a right angle traingle ; ∠ABO = 90°

➠ According to Pythagorus Theorem,

$\implies \boxed{\sf \: Hypotenuse^{2} = {Base}^{2} + {Height}^{2} }$

$\implies \sf \: OA^{2} = OB^{2} + AB^{2}$

$\implies \sf \: {70}^{2} = {r}^{2} + {56}^{2}$

$\implies \sf \: {r}^{2} = {70}^{2} - {56}^{2}$

$\implies \sf \: {r}^{2} = 4900 - 3136$

$\implies \sf \: {r}^{2} = 1764$

$\implies \sf \boxed{ \red{ \bold{r = 42}}}$

➦ Now, the circumference of circle,

$\implies \boxed{ \sf \: C = 2\pi r}$

$\implies \sf \: C = 2 \times \dfrac{22}{7} \times 42$

$\implies \sf \: C = 2 \times 22 \times 6$

$\implies \sf \boxed{ \red{ \bold{ C = 264 \: cm}}}$

Answer 2

● See the attachment diagram.

$\Large\bf\pink{GiVeN,}$

• $\bf{\red{Length\:of\:tangent}}\:(AP)\:=\:56\:cm\:$

• $\bf{\red{Length\:of\:pt.\:A\:from\:the\:centre\:\atop {of\:the\:circle}}\:(OA)\:=\:70\:cm}\:$

$\Large\bf\purple{FoRmUlA,}$

▪To calculate radius of the circle,

$\green\bigstar\:\:\bf\blue{Radius\:(OP)\:=\:\sqrt{(OA)^2\:-\:(AP)^2}\:}$

$\Large\bf\orange{CaLcUlAtIoN,}$

$:\implies\:\:\bf{Radius\:(OP)\:=\:\sqrt{(70)^2\:-\:(56)^2}\:}$

$:\implies\:\:\bf{Radius\:(OP)\:=\:\sqrt{4900\:-\:3136}\:}$

$:\implies\:\:\bf{Radius\:(OP)\:=\:\sqrt{1764}\:}$

$:\implies\:\:\bf\green{Radius\:(OP)\:=\:42\:cm\:}$

▪To calculate circumference of the circle,

$\red\bigstar\:\:\bf\orange{Circumference \:=\:2\pi {r}\:}$

$\longmapsto\:\:\bf{Circumference \:=\:2\times {\dfrac{22}{7}}\times {42}\:}$

$\longmapsto\:\:\bf{Circumference \:=\:2\times{22}\times {6}\:}$

$\longmapsto\:\:{\boxed {\green{\bf {Circumference \:=\:264\:cm^2\:}}}}$