Question

If arcs of same length in two circles subtend angles ${{60}^{\circ }}$ and ${{75}^{\circ }}$ at their centers, find the ratios of their radii.

Answer 1

Hint:Here in this question first we have to change angles in degree to radian by multiplying degree with $\dfrac{\pi}{180}$, then use the formula $s=r\theta$ to find the arc length of both the circles individually then equate it with the arc lengths of both subtended angles as given in question that they subtend at same arc lengths simplify it and find ratio of radius of circles.

Complete step-by-step answer:
In the question we are given a condition that two areas of same length in to circles subtend angles ${{60}^{\circ }}$ and ${{75}^{\circ }}$at their centres so now we are asked to find the ratios of radii.
Before proceeding we will first briefly say something about radian.
The radian is an S.I. unit for measuring angles and is the standard unit of angular measure used in areas of mathematics. The length of an arc of a unit circle is numerically equal to the measurement in radians of the angle that it subtends; one radian is just under 57.3 degrees.
Radian describes the plain angle subtended by a circular arc as the length of arc divided by radius of the arc. One radian is the angle subtended at the center of a circle by an arc that is equal in length to the magnitude in radians of such a subtend angle is equal to the ratio of the arc length to the radius of circle; that is $\theta \ =\ \dfrac{s}{r}$, where $\theta$ is the subtended angle in radians, s is arc length and r is radius .
Conversely, the length of the enclosed arc is equal to the radius multiplied by the magnitude of the angle in radians that is $s=r\theta$.
So, let the circle with arc length $s_1$, subtend ${{60}^{\circ }}$ and radius let’s suppose it is $r_1$.
So, we can use formula $s=r\theta$. Where s is length, r is radius of circle $\theta$ be angle in radian.
If $\theta$ is in degree so we will convert it by multiplying it by $\dfrac{\pi }{180}$.
We are given angle is ${{60}^{\circ }}$ so in radian it will be $60\times \dfrac{\pi }{180}$ or $\dfrac{\pi }{3}$.
So, on applying formula $s=r\theta$ we get,
${{s}_{1}}\ =\ \dfrac{\pi }{3}\times {{r}_{1}}\ =\ \dfrac{\pi {{r}_{1}}}{3}...........(1)$
Now let the circle with arc length the $s_2$ subtend ${{75}^{\circ }}$ and radius let’s suppose be $r_2$.
So, we can use the formula $s=r\theta$, where s is length, r is radius of circle and $\theta$ be angle in radian.
If $\theta$ is in degree so we will convert it by multiplying it by $\dfrac{\pi }{180}$.
We are given angle is ${{75}^{\circ }}$ so, in radian it will be $75\times \dfrac{\pi }{180}$ or $\dfrac{5\pi }{12}$.
So on applying formula $s=r\theta$ we get,
${{s}_{2}}\ =\ \dfrac{5\pi }{12}\times {{r}_{2}}\ =\ \dfrac{5\pi {{r}_{2}}}{12}..........(2)$
In question given that both angles subtend at same arc length So, Equating equation (1) and (2), we get
$s_1=s_2$
$\dfrac{\pi {{r}_{1}}}{3}\ =\ \dfrac{5\pi {{r}_{2}}}{12}$
So, on cross multiplication we get,
$4{{r}_{1}}\ =\ 5{{r}_{2}}$
Hence, we can write $\dfrac{{{r}_{1}}}{{{r}_{2}}}\ =\ \dfrac{5}{4}$.
So, the ratio is $5:4$.

Note: For finding the ratio one can also use formula to find an arc by calculating using expression $2\pi r\left( \dfrac{\theta }{360} \right)$ where $\theta$ is in degree instead of changing it into radian.Students should remember to convert from degree to radian one should multiply by $\dfrac{\pi }{180}$ to get the value in radians and to convert from radian to degree one should multiply by $\dfrac{180 }{\pi}$ to get the value in degrees.