Let A and B be two points on a circle with centre O. Are there points X, Y on the circle Class 9

Let A and B be two points on a circle with centre O. Are there points X, Y on the circle Class 9

Question 1.

Let A and B be two points on a circle with centre O. Class 9

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?

Solution:

i.

Let A and B be two fixed points on a circle.

Take any two points X and Y on the same side of AB on the circle.

∵ Since angles subtended by the same arc in the same segment of a circle are equal.

∴ ∠AXB = ∠AYB

∴ It is not possible to find such points X and Y where ∠AXB ≠ ∠AYB.

ii. No, not always if ∠AXB = ∠AYB = 90° then X and Y may lie on opposite sides of AB.

iii. If ∠AXB = ∠AYB, then circle passing through A, B, and X will also pass through Y.

By converse of theorem "angles in same segment of circle", if a line segment joining two point subtends equal angles at two other points on same side of line segment, then the four points lie on a circle, i.e., A, X, Y and B are concyclic.