Class: 10th | Mathematics (FBISE) | Lecture # | Unit # 12 | Theorem # 1 | Central angles | скачать в хорошем качестве

Class: 10th | Mathematics (FBISE) | Lecture # | Unit # 12 | Theorem # 1 | Central angles |

Class: 10th | Mathematics (FBISE) | Lecture # | Unit # 12 | Theorem # 1 | Central angles | Mathematics Science Group | The measure of a central angle of a minor arc of a circle, is double that of the angle subtended by the corresponding major arc | The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle | Dear viewers, it is my pleasure to deliver you mathematics tutorials in simple and native language so that you can get it easily | #MathsMadeEasy is a channel where you can improve your #Mathematics | This is an education channel where maths made easy will try to solve your problems | Students may send the problems they are facing through comments | Prove that the angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle . Given : An arc PQ of a circle subtending angles POQ at the centre O and PAQ at a point A on the remaining part of the circle. To prove : ∠POQ=2∠PAQ To prove this theorem we consider the arc AB in three different situations, minor arc AB, major arc AB and semi-circle AB. Construction : Join the line AO extended to B. Proof : ∠BOQ=∠OAQ+∠AQO .....(1) Also, in △ OAQ, OA=OQ [Radii of a circle] Therefore, ∠OAQ=∠OQA [Angles opposite to equal sides are equal] ∠BOQ=2∠OAQ .......(2) Similarly, BOP=2∠OAP ........(3) Adding 2 & 3, we get, ∠BOP+∠BOQ=2(∠OAP+∠OAQ) ∠POQ=2∠PAQ .......(4) For the case 3, where PQ is the major arc, equation 4 is replaced by Reflex angle, ∠POQ=2∠PAQ Note that: The angle subtended at the centre is the angle subtended by an arc or a chord or a sector at the centre of a circle. It is referred to as the central angle. Just visualise a V drawn from the ends of a chord to the centre. It has its vertex at the center of the circle, and its sides are radii of the circle. Similarly the angle subtended at the circles is the angle made by a chord to the circle. Just visualise a V drawn from ends of a chord to a point on the circle. So two Vs can be visualised from a chord, one to centre, one to the circle. The angle subtended at the centre is double the angle subtended at the circle by the arc, chord, sector. Those two profound facts of circles, logically leads to several other facts. The diameter is the biggest chord and infinite diameters are possible. In such cases, the central angle is 180° and the angle subtended at the circle is 90°. So logically, the angle subtended at the semicircle is right angle, that's 90°. The central angle will be lss than180° in a minor arc and greater than180° (reflex angle) in a major arc. The central angle around the centre is 360° and so the sum of opposite angles off cyclic quadrilateral will be supplementary, that's 180°. Logically if one angle in a cyclic quadrilateral is acute, the other will be obtuse such that they are supplement to it. Only in cyclic squares and rectangles the opposite angles will be right angles, that's 90° each.