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Prove the theorem of alternate segment |
| Answer» Alternate Segment:If a chord is drawn through the point of contact of a tangent to a circle, then the angles which this chord makes with the given tangent are equal respectively to the angles formed in the corresponding alternate segments.Given : is the tangent to the circle with centre O. AB is the chord drawn at the point of contact A.C and D are two points on the circumference such that they lie on either side of the chord.To Prove: (i) and (ii) Construction: Draw the diameter AOE and draw EB.Proof: In the figure, (radius is perpendicular to the tangent at the point of contact) (1)In DEAB (Angle in a semi-circle) (2)From (1) and (2)But, (Angles in the same segment)Now, consider cyclic quadrilateral ACBD. (Opposite angles) .(4)And (Linear pair)But (5)From (4) and (5)Hence proved. | |