 Hi and welcome to the session. I am Asha and I am going to help you with the following question which says If in two circles arcs of same length sub 10 angles 60 degree and 75 degree at the centre find the ratio of their radii So let us proceed on with the solution And here we are given two circles In both of these circles we have two arcs which are of same length and one sub 10 an angle of 60 degree at the centre And another arc sub 10 an angle of 75 degree at the centre And length of both these arcs AB and CTR equal, let them be equal to x centimetre And let O be the centre first circle and O dash be the centre of another circle So we have to find the ratio of their radii Now let R be the radius of first circle and R dash be the radius of another circle Let this be first circle and this be second circle Now as we know length of arc of a circle is equal to 2 pi arc into theta upon 360 degree But theta is the angle subtended by the arc at the centre So in the first circle we are applying this formula to x Which is the length of arc into 2 into pi into arc And theta in the first circle is 60 degrees 360 So here we have 0, here we have 6, 6 and that is x 2 into 3 is 6 We have x is equal to pi R upon 3 Which implies R is equal to 3x upon pi Now let us consider the second circle and we are applying this formula Here length of the arc is x since arc of both the circles are same And we have 2 pi radius, here r was the radius And the second circle radius we have considered as R dash Then we have theta Theta in the second circle is 75 degrees upon 360 Which implies R dash is equal to 360 into x upon 2 pi into 75 degree Now 2 is the common factor of the numerator and denominator So 2 is 2, 2 into 1 is 360 And now 5 is the common factor, 5 into 15 is 75 And 5 3 is 15, 5 3 is 30 Again 3 is the common factor, so we have 3 5 is 15, 3 12 is 36 So this implies R dash is equal to 12x upon 5 pi Now according to the question we have to find the ratio of their radius So what we will do is, divide the radius of second circle And the radius of first circle Radius of first circle upon radius of second circle This is what we have to find And radius of the circle, small r, radius of second circle is R dash The small r we will do 3x upon pi And small r dash is 12x upon 5 pi Which is further equal to 3x upon pi into 5 pi upon 12x Pi cancels out with pi into x And we have 3 4s at 12 This gives 5 upon 4 Hence, ratio of r and r dash is equal to 5 is to 4 So the answer is 5 is to 4 is the ratio of the radius So this completes the solution Hope you enjoyed it Take care and have a...