1. Day 132 – Similarity between radius and length of an arc

2. Introduction
Similarity is a concept used to compare two quantities or entities that are proportional. We have used it in triangles and quadrilaterals to proof quite a number of theorems. However, that is not enough, it is still valid to set some concepts on circles straight. Our greatest interest is to see how it is used to bring about the relationship of angles and arch lengths. In this lesson, we are going to derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius

3. Vocabulary
Radius
The distance from the center of a circle to its circumference
Arc length
It is the distance between two points on an curved line.

4. We claim that The length of the arc intercepted by an angle is proportional to the radius Consider a circle and an arc of the same radius. Let the circumference of the circle be 𝑐 and the length of the arc be 𝑙. Thus there central angles are 360°= 360°× 𝜋 180 =2𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 and 𝜃 respectively. 2𝜋 𝑐 𝑟 𝜃 𝑙 𝑟

5. When sector is translated to the circle such that they share the center, the arc of the sector will lie of the circumference of the circle since they have the same radius. Thus, we can get a suitable scale factor that can be used to dilate the arc to be equal to the circumference of the circle. Therefore, the two, the circle and the sector, are similar, consequently, their arc’s. Since the central angle determines the length of the arc, we have the similarity relation 𝑐 2𝜋 = 𝑙 𝜃

6. By definition, 𝑐=2𝜋𝑟. Upon substitution, we get 2𝜋𝑟 2𝜋 = 𝑙 𝜃 𝑟= 𝑙 𝜃 Thus, 𝑙=𝑟𝜃 Since, 𝜃 is a constant ( in radians), we have the proportionality relation 𝑙∝𝑟 and the constant of proportionality, 𝑘=𝜃 (given in radians).

7. Example A sector or radius 7 𝑖𝑛 has a central angle of 80°, find the relationship between the length of its arc and the radius. Solution The sector is similar to the circle whose central angle is 2𝜋. The central angle of the arc is 80°. In radian, we multiply it by 𝜋 180 to get 80× 𝜋 180 = 4 9 𝜋 The proportional relation is 𝑐 2𝜋 = 𝑙 4𝜋/9 Since 𝑐=2𝜋𝑟, we have

8. 2𝜋𝑟 2𝜋 = 𝑙 4𝜋/9 𝑟= 𝑙 4𝜋/9 hence 𝑙= 4𝜋 9 𝑟

9. homework
A sector or radius 12 𝑖𝑛 has a central angle of 45°, find the relationship between the length of its arc and the radius.

10. Answers to homework
𝑙= 𝜋 4 𝑟

11. THE END