Question

If in two circles, arcs of the same length subtend angles 60° and 75° at the centre, find the ratio of their radii.


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Answer 1

Answer:

length of an arc s= radius *angle

length of two circles are equal, then

s= radius of 1st circle *angle of 1st circle = radius of 2nd circle* angle of 2nd circle

r* 60 =r* 75

r/r = 75/60

r/r =5/4

the ratio of their radii =5:4

Answer 2

★ Concept :-

Here the concept of Radians has been used. We see that we are given the angle subtended by each arc with the centre. Also we see that these angles are in degrees. So firstly we need to convert each angle into radians. Then we can use the relationship between arc and radius. This can give us the ratio of their radii.

Let's do it !!

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★ Formula Used :-

$\;\;\boxed{\sf{\pink{Radian\;Measure\;=\;\dfrac{\pi}{180^{\circ}}\:\times\:Degree\;Measure}}}$

$\;\;\boxed{\sf{\pink{\theta_{in\:radians}\;=\;\dfrac{Arc\:(l)}{Radius\:(R)}}}}$

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★ Solution :-

Given,

» Angle subtended by first arc in first circle = 60°

» Angle subtended by second arc in second circle = 75°

It's given that the length of arcs of both circle is same. So,

• Let the length of arc for both circle be 'l'

Now let's assume for radius.

• Let the radius of first circle be R₁

• Let the radius of second circle be R₂

• Let the angle subtended in first circle in radians be θ₁

• Let the angle subtended in second circle in radians be θ₂

• For the radian measure of each angle ::

We have the formula that,

$\;\;\tt{\rightarrow\;\;Radian\;Measure\;=\;\dfrac{\pi}{180^{\circ}}\:\times\:Degree\;Measure}$

Now we shall apply values here to get answer.

Let's do it for θ₁ firstly. Then,

$\;\;\tt{\rightarrow\;\;\theta_{1}\;=\;\dfrac{\pi}{180^{\circ}}\:\times\:60^{\circ}}$

$\;\;\tt{\rightarrow\;\;\theta_{1}\;=\;\dfrac{\pi}{3}\:\times\:1}$

$\;\;\bf{\rightarrow\;\;\orange{\theta_{1}\;=\;\dfrac{\pi}{3}}}$

Let's now calculate the value of θ₂ . Then,

$\;\;\tt{\rightarrow\;\;\theta_{2}\;=\;\dfrac{\pi}{180^{\circ}}\:\times\:75^{\circ}}$

$\;\;\tt{\rightarrow\;\;\theta_{2}\;=\;\dfrac{\pi}{12}\:\times\:5}$

$\;\;\bf{\rightarrow\;\;\blue{\theta_{2}\;=\;\dfrac{5\pi}{12}}}$

• For the length of arc for both angles ::

We have the formula that,

$\;\;\tt{\Longrightarrow\;\;\theta_{in\:radians}\;=\;\dfrac{Arc\:(l)}{Radius\:(R)}}$

Now let's calculate it for θ₁ . Then,

$\;\;\tt{\Longrightarrow\;\;\theta_{1}\;=\;\dfrac{l}{R_{1}}}$

By applying values, we get

$\;\;\tt{\Longrightarrow\;\;\dfrac{\pi}{3}\;=\;\dfrac{l}{R_{1}}}$

$\;\;\bf{\Longrightarrow\;\;\orange{l\;=\;\dfrac{R_{1}\pi}{3}}}$

Now let's calculate it for θ₂ . Then,

$\;\;\tt{\Longrightarrow\;\;\theta_{2}\;=\;\dfrac{l}{R_{2}}}$

By applying values, we get

$\;\;\tt{\Longrightarrow\;\;\dfrac{5\pi}{12}\;=\;\dfrac{l}{R_{2}}}$

$\;\;\bf{\Longrightarrow\;\;\blue{l\;=\;\dfrac{5R_{2}\pi}{12}}}$

(Length of arc for both circle is same)

• For the Ratio of Radii :::

We see that l is same for both circles. So equating the values of l, we get

$\;\tt{\rightarrowtail\;\;\dfrac{R_{1}}{R_{2}}\;=\;\dfrac{5R_{2}\pi}{12}}$

In order to find ratio, we need to bring both radii to same side. Then,

$\;\tt{\rightarrowtail\;\;\dfrac{R_{1}}{R_{2}}\;=\;\dfrac{5\pi\:\times\:3}{12\pi}}$

Now cancelling π from numerator and denominator, we get

$\;\tt{\rightarrowtail\;\;\dfrac{R_{1}}{R_{2}}\;=\;\dfrac{5\times\:3}{12}}$

Now cancelling the related terms, we get

$\;\tt{\rightarrowtail\;\;\dfrac{R_{1}}{R_{2}}\;=\;\dfrac{5\times\:1}{4}}$

$\;\tt{\rightarrowtail\;\;\red{\dfrac{R_{1}}{R_{2}}\;=\;\dfrac{5}{4}}}$

$\;\tt{\rightarrowtail\;\;\green{R_{1}\;:\;{R_{2}\;=\;5\;:\;4}}}$

This is the required answer.

$\;\underline{\boxed{\tt{Required\;\:ratio\;\:of\;\:radii\;=\;\bf{\purple{5\;:\;4}}}}}$

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★ More to know :-

$\;\;\sf{\leadsto\;\;Degree\;Measure\;=\;\dfrac{180^{\circ}}{\pi}\:\times\:Radian\;Measure}$

$\;\;\sf{\leadsto\;\;Area\;of\;Sector\;=\;\dfrac{\pi r^{2}\theta}{360^{\circ}}}$

$\;\;\sf{\leadsto\;\;Length\;of\;Arc\;=\;\dfrac{2\pi r\theta}{360^{\circ}}}$

• sin² x + cos² x = 1

• sec² x = 1 + tan² x

• cosec² x = 1 + cot² x