Question

If in two circles, arcs of the same length subtend angles \[{60^0}\] and \[{75^0}\] at the Centre, find the ratio of their radii.

Answer 1

Hint:
As we know that the arc length for a circle which subtend the angle x at Centre is given by, \[l = rx\] where x is given in radian and so as given that the above two given arc lengths are same so put the given angles in radians and equate them. Hence, from there we can obtain the ratio oF radii of both the circles.

Complete step by step solution:

As we know that arcs lengths are same and subtends angles \[{60^0}\] and \[{75^0}\] at the Centre

As we know that arc length \[l = r\theta \]
Converting the given angles into radian as,
For \[{60^0}\] ,
\[\theta = 60 \times \dfrac{\pi }{{180}} = \dfrac{\pi }{3}\]
Similarly, for \[{75^0}\] we get,
\[\theta = 75 \times \dfrac{\pi }{{180}} = \dfrac{{5\pi }}{{12}}\]
Now as arc lengths are equal so we get,
\[{l_1} = {l_2}\]
On substituting the above equation we get,
\[{r_1}{\theta _1} = {r_2}{\theta _2}\]
Now, as we need the ratio of the radii so we get,
\[\dfrac{{{r_1}}}{{{r_2}}} = \dfrac{{{\theta _2}}}{{{\theta _1}}}\]
On putting the value of above obtained angle in radians we get,
\[\dfrac{{{r_1}}}{{{r_2}}} = \dfrac{{\dfrac{{5\pi }}{{12}}}}{{\dfrac{\pi }{3}}}\]
Hence, on simplifying above equation we get,
\[\dfrac{{{r_1}}}{{{r_2}}} = \dfrac{{\dfrac{5}{{12}}}}{{\dfrac{1}{3}}}\]
On calculating above ratio we get,
\[\dfrac{{{r_1}}}{{{r_2}}} = \dfrac{5}{4}\]
Hence, above is our required answer.

Note:
Arc length is the distance between two points along a section of a curve. Determining the length of an irregular arc segment is also called rectification of a curve. The advent of infinitesimal calculus led to a general formula that provides closed-form solutions in some cases. Substitute the value and solve the above equation carefully so that without any mistake the correct answer can be calculated.