Ex 5.6, 2

Transcript

Ex 5.6, 2 Let A and B be two points on a circle with centre O. (i) Are there points X,Y on the circle, on the same side of AB, such that ∠AXB is different from ∠AYB? (ii) Is it true that if ∠AXB=∠AYB, then X and Y lie on the same side of the circle? (iii) If ∠AXB=∠AYB, and X and Y do not lie on the circle, does the circle through A,B and X also pass through Y? Let’s draw the diagram We answer the questions one by one (i) Are there points X,Y on the circle, on the same side of AB, such that ∠AXB is different from ∠AYB? We know that Angles in same segment are equal Since both ∠ AXB and ∠ AYB are in same arc segment AB ∴ ∠ AXB = ∠ AYB Thus, the answer is no. Because if X & Y are in same side of AB, then ∠𝐀𝐗𝐁 =∠𝐀𝐘𝐁 (ii) Is it true that if ∠AXB=∠AYB, then X and Y lie on the same side of the circle? It is not always true Because X & Y can be on opposite arcs and be equal also (Eg: if AB is the diameter, than ∠ AXB = ∠ AYB = 90° - and X & Y can be on oppsite sides of AB) But, if they are on same segment – they are always equal (iii) If ∠AXB=∠AYB, and X and Y do not lie on the circle, does the circle through A,B and X also pass through Y? We have two cases here When X & Y are on same side of AB When X & Y are on not on same side of AB (iii) If ∠AXB=∠AYB, and X and Y do not lie on the circle, does the circle through A,B and X also pass through Y? We have two cases here When X & Y are on same side of AB When X & Y are on not on same side of AB This question tests the Converse of Angles in the Same Segment Theorem, which states: If a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the segment, the four points are concyclic (they lie on the same circle). Now, If X and Y are on the same side of AB, then yes, the circle passing through A, B, and X will definitively pass through Y. However, if X and Y are on opposite sides of AB, they will generally not lie on the same circle (unless both angles happen to be exactly 90°). Since the question does not specify their position relative to the line AB, we cannot conclude it is always true. Now, our question was If ∠AXB=∠AYB, and X and Y do not lie on the circle, does the circle through A,B and X also pass through Y Thus, our answer is Not necessarily. It is only guaranteed if X and Y are on the SAME side of line AB.