Theorem 1 Angles in the same segment of a circle are equal.
Given A circle C(O,r), an arc PQ and two angles in the same segment of the circle. To prove . Construction Join OP and OQ. PROOF We know that the angle subtended by an arc at the centre is double the angle subtended by the arc at any point in the remaining part of the circle . So, in figure, we have
Hence proved. |
Converse Theorem: If a line segmentjoining two points subtends equal angles at two other points lying on the same side of the line containing the line segment. the four points lie on the circle. | |
Given: A line segment AB and two points C and D such that To Prove: The points A, B, C and D lie on a circle or the points are concyclic Construction: Draw a circle Which passes through A, B C Proof As the circle passes through A, B and C, let us assume that it does not pass through D. Then either the circle will intersect AD at F or will cut the line segment AD extended to E Case I: THe circle intersects AD at F . Then A,C,F,B lie on the circle ....... 1 [ Angle in the same segment are equal] but ........2 [ Given] From Equation 1 and 2 we get This is not possible because is an exterior angle to and we know that exterior angle of a triangle can not be equal to interior opposite angle. Hence our Assumption is wrong. Therfore D and F should coincide Case 2: The circle cuts the line segment AD extended to E Then A,C,E,B lie on the circle ....... 3 [ Angle in the same segment are equal] but ........4 [ Given] From Equation 3 and 4 we get This is not possible because is an exterior angle to and we know that exterior angle of a triangle can not be equal to interior opposite angle. Hence our Assumption is wrong. Therfore D and E should coincide Hence the circle has to pass through A, B, C and C or These points are concyclic | |
Illustration: If in the figure Solution: In the figure we have a circle with center O and AC is the arc The arc AC subtends angle at point B and D on the circle Now are in the same segment because they are on the same side of the arc Therefore [Angles in the same segment are equal] As | ​​ |
| |||||
Right Option : B | |||||
View Explanation |
If AB is a chord of a circle, P and Q are the points on the circle different from A and B, then | |||
Right Option : B | |||
View Explanation |
| |||||
Right Option : B | |||||
View Explanation |
It was good as the experience because as we had come here we had been improved in a such envirnment created here.Extra is taught which is beneficial for future.
Abhyas is a complete education Institute. Here extreme care is taken by teacher with the help of regular exam. Extra classes also conducted by the institute, if the student is weak.
My experience was very good with Abhyas academy. I am studying here from 6th class and I am satisfied by its results in my life. I improved a lot here ahead of school syllabus.
I have spent a wonderful time in Abhyas academy. It has made my reasoning more apt, English more stronger and Maths an interesting subject for me. It has given me a habbit of self studying
It was a good experience with Abhyas Academy. I even faced problems in starting but slowly and steadily overcomed. Especially reasoning classes helped me a lot.
One of the best institutes to develope a child interest in studies.Provides SST and English knowledge also unlike other institutes. Teachers are co operative and friendly online tests andPPT develope practical knowledge also.
It has a great methodology. Students here can get analysis to their test quickly.We can learn easily through PPTs and the testing methods are good. We know that where we have to practice
Being a parent, I saw my daughter improvement in her studies by seeing a good result in all day to day compititive exam TMO, NSO, IEO etc and as well as studies. I have got a fruitful result from my daughter.
My experience with Abhyas academy is very good. I did not think that my every subject coming here will be so strong. The main thing is that the online tests had made me learn here more things.
My experience with Abhyas is very good. I have learnt many things here like vedic maths and reasoning also. Teachers here first take our doubts and then there are assignments to verify our weak points.