Or, Find the sum of all two-digit odd posive lhul17. In the adjoining

1.	Or, Find the sum of all two-digit odd posive lhul17. In the adjoining figure, PA and PB aretangents drawn from an external point P to acircle with centre OProve that ZAPB- 2ZOAB
Answer» given : a circle with centre O. PA and PB are two tangents to the circle at A and B To prove : L APB = 2 <OAB Proof : join AB and OA OA is perpendicular to PA (tangent and the radius at the point of contact) therefore <OAP = 90degree let <OAB = x therefore <PAB = 90 - x ----- (1) PA = PB (tangent from same external point) therefore triangle APB is isosceles therefore <PAB = < PBA = 90-x (angles opp to equal sides) Tn triangle APB <P + 90 - x + 90 - x = 180 degree (angle sum property) <P + 180 - 2x = 180degree <P = 2x < APB = 2x ie, < APB = 2 < OAB

Or, Find the sum of all two-digit odd posive lhul17. In the adjoining figure, PA and PB aretangents drawn from an external point P to acircle with centre OProve that ZAPB- 2ZOAB

Answer»

given : a circle with centre O.

PA and PB are two tangents to the circle at A and B

To prove : L APB = 2 <OAB

Proof : join AB and OA

OA is perpendicular to PA (tangent and the radius at the point of contact)

therefore <OAP = 90degree

let <OAB = x

therefore <PAB = 90 - x ----- (1)

PA = PB (tangent from same external point)

therefore triangle APB is isosceles

therefore <PAB = < PBA = 90-x (angles opp to equal sides)

Tn triangle APB

<P + 90 - x + 90 - x = 180 degree (angle sum property)

<P + 180 - 2x = 180degree

<P = 2x

< APB = 2x

ie, < APB = 2 < OAB