If Pa and Pb Are Tangents from an Outside Point P. Such that Pa = 10 Cm and ∠Apb = 60°. Find the Length of Chord Ab. - Mathematics

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If PA and PB are tangents from an outside point P. such that PA = 10 cm and ∠APB = 60°. Find the length of chord AB.

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Solution

AP = 10 cm ∠APB = 60°

Represented in the figure

We know that

A line drawn from center to point from where external tangents are drawn divides or

bisects the angle made by tangents at that point ∠APO = ∠OPB =`1/2`× 60° = 30°

The chord AB will be bisected perpendicularly

∴ AB = 2AM

In ΔAMP,

sin 30° =`"𝑜𝑝𝑝.𝑠𝑖𝑑𝑒"/"ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒"="(AM)/(AP)`

AM = AP sin 30°

=`(AP)/2 =10/2`= 5𝑐𝑚

AP = 2 AM = 10 cm                      ---- Method (i)

In ΔAMP, ∠AMP = 90°, ∠APM = 30°

∠AMP + ∠APM + ∠MAP = 180°

90° + 30° + ∠MAP = 180°

∠MAP = 180°

In ΔPAB, ∠MAP = ∠BAP = 60°, ∠APB = 60°

We also get, ∠PBA = 60°

∴ΔPAB is equilateral triangle

AB = AP = 10 cm.                  -----Method (ii)

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Chapter 8: Circles - Exercise 8.2 [Page 34]

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RD Sharma Class 10 Maths
Chapter 8 Circles
Exercise 8.2 | Q 12 | Page 34

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