Question 29 (Choice 1) - CBSE Class 10 Sample Paper for 2024 Boards - Maths Standard - Solutions of Sample Papers for Class 10 Boards

Last updated at Aug. 4, 2023 by Teachoo

PA and PB are tangents drawn to a circle of centre O from an external point P. Chord AB makes an angle of 30° with the radius at the point of contact. If length of the chord is 6 cm, find the length of the tangent PA and the length of the radius OA

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PA and PB are tangents drawn to a circle of centre O from an external point P. Chord AB makes an angle of 30° with the radius at the point of contact. If length of the chord is 6 cm, find the length of the tangent PA and the length of the radius OA ∠PAB = 90° − 30 °
∴ ∠PAB = 60°
Also, since
Tangents from external point are equal
PA = PB
And we know angles opposite to equal sides are equal
Thus, ∠ PAB = ∠ PBA = 60°
In Δ APB
By Angle Sum property
∠ APB + ∠ PAB + ∠ PBA = 180°
∠ APB + 60° + 60° = 180°
∠ APB + 120° = 180°
∠ APB = 180° − 120°
∠ APB = 60°
Thus, in Δ APB
∠ PAB = ∠ PBA ∠ APB = 60°
Since all angles are 60°,
∴ Δ APB is an equilateral triangle
And, in equilateral triangle, all sides are equal
∴ PA = AB
PA = 6 cm
Finding radius OA
Since ∠APB = 60°
By symmetry
∠ OPA = (∠ 𝐴𝑃𝐵)/2 = (60° )/2
∠ OPA = 30°
Using Trigonometry in right Δ AOP
tan P = (𝑆𝑖𝑑𝑒 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑡𝑜 𝑎𝑛𝑔𝑙𝑒" " 𝑃)/█(𝑆𝑖𝑑𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 𝑎𝑛𝑔𝑙𝑒" " 𝑃@ )
tan 30° = 𝑶𝑨/𝑷𝑨
1/√3 = 𝑂𝐴/6
0A = 𝟔/√𝟑
OA = (2 × 3)/√3
OA = 2√𝟑 cm

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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