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प्रश्न
In the following figure, ∠AOB = 90º and ∠ABC = 30º, then ∠CAO is equal to ______.
विकल्प
30º
45º
90º
60º
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उत्तर
In the following figure, ∠AOB = 90º and ∠ABC = 30º, then ∠CAO is equal to 60º.
Explanation:
Since, the angle subtended by an arc at the centre is twice the angle subtended by it at any point on the remaining part of the circle.
∴ ∠AOB = 2∠ACB
`\implies` 90° = 2∠ACB ...[∵ ∠AOB = 90°]
`\implies` ∠ACB = 45°
Also, AO = OB ...[Radii of the same circle]
`\implies` ∠ABO = ∠BAO ...(i) [Angles opposite to equal sides are equal]
Now, in ΔOAB, ∠OAB + ∠ABO + ∠BOA = 180° ...[Sum of angles of a triangle is 180°]
∴ ∠OAB + ∠OAB + 90° = 180° ...[From (i)]
`\implies` 2∠OAB = 180° – 90° = 90°
`\implies ∠OAB = 90^circ/2 = 45^circ` ...(ii)
Also, in ΔACB, ∠ACB + ∠CBA + ∠CAB = 180° ...[Sum of angles of a triangle is 180°]
∴ 45° + 30° + ∠CAB = 180° ...[∵ ∠ABC = 30°]
`\implies` ∠CAB = 105°
Since, ∠CAO + ∠OAB = ∠CAB
`\implies` ∠CAO + 45° = 105° ...[From (ii)]
`\implies` ∠CAO = 105° – 45° = 60°
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