AG

- 00:00
- Let's learn about a very interesting circles theorem!!!!
- 00:04
- The angle subtended by an arc of a circle at its center is twice the angle it subtends anywhere on the circle’s circumference.
- 00:20
- Consider the following figure, in which an arc (or segment) AB subtends ∠AOB at the center O and ∠ACB at a point C on the circumference.
- 00:38
- We have to prove that ∠AOB = 2 × ∠ACB. Draw the line through O and C, and let it intersect the circle again at D, as shown.
- 00:54
- There are two triangles formed ΔOAC and ΔOBC. So, we make the following observations. 1. In ΔOAC, ∠OAC = ∠OCA, because OA = OC 2. In ΔOBC, ∠OBC = ∠OCB, because OB = OC. Hence, using the exterior angle theorem, we get, ∠AOD= 2×∠ACO ⋯ (1) ∠DOB= 2×∠OCB ⋯ (2) Add equations (1) and (2): ∠AOD+ ∠DOB= 2× (∠ACO+ ∠OCB) ⇒ ∠AOB= 2× ∠ACB
- 01:38
- I learnt the theorem... That was so easy!!!
- 01:42
- M a t h e m a t i c s A r o u n d U s: PRACTICAL APPLICATION OF THIS THEOREM
- 01:44
- CAMERA: determine the focal length in camera lens VIDEO GAMES :to navigate around objects in video games SCIENCE: The Large Hadron Collider in Europe is a tunnel in the shape of a circle which helps force the particles to move. ARCHITECHTURE: Domes, decorative features in buildings and circular-shaped windows