Given two central angles in any two circles, the ratio of the arc length to the radius of each corresponding circle is equal if and only if each central angle is equal. That is, given central angle θ1 degrees with arc length s1 in a circle of radius r1 and central angle θ2 degrees with arc length s2 in a circle of radius r2, then s1/r1=s2/r2 if and only if θ1 = θ2. Prove the above theorem using the fact that the ratio of an arc length over the circumference of a circle is the same as the ratio of its central angle (in degrees) over 360 degrees. Hint: First write the above proportion for each circle. Then apply it to s1/r1=s2/r2 and conclude that θ1 = θ2.

Question

Cristal Maynard

Mathematics

24 November, 15:27

Given two central angles in any two circles, the ratio of the arc length to the radius of each corresponding circle is equal if and only if each central angle is equal. That is, given central angle θ1 degrees with arc length s1 in a circle of radius r1 and central angle θ2 degrees with arc length s2 in a circle of radius r2, then s1/r1=s2/r2 if and only if θ1 = θ2. Prove the above theorem using the fact that the ratio of an arc length over the circumference of a circle is the same as the ratio of its central angle (in degrees) over 360 degrees. Hint: First write the above proportion for each circle. Then apply it to s1/r1=s2/r2 and conclude that θ1 = θ2.

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Answers (1)

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Adam Hatfield · Answer 1

First, we have

s1/r1 = s2/r2

The question also states the fact that

s/2πr = θ/360°

Rearranging the second equation, we have

s/r = 2πθ/360°

Then we substitute it to the first equation

s1/r1 = 2πθ1/360°

s2/r2 = 2πθ2/360°

which is now

2πθ1/360° = 2πθ2/360°

By equating both sides, 2π and 360° will be cancelled, therefore leaving

θ1 = θ2