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Theorem 10.8 : The angle subtended by an...

Theorem 10.8 : The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.

Text Solution

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In `/_\APO`
`OP=OA` (radius)
`/_OPA=/_OAP` (angle opposite to eqal sides are equal)
By exterior angle property,
`/_BOP=/_OPA+/_OAP`
`/_BOP=2/_OAP`
Similarly, in `/_\AQO`
...
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The angle subtended by an arc of a circle at the centre is double the angle subtended by it any point on the remaining part of the circle.

Show that any angle in a semi-circle is a right angle. The following arc the steps involved in showing the above result. Arrange them in sequential order. A) therefore angleACB=180^(@)/2=90^(@) B) The angle subtended by an arc at the center is double of the angle subtended by the same arc at any point on the remaining part of the circle. c) Let AB be a diameter of a circle with center D and C be any point on the circle. Join AC and BC. D) therefore angleAD = 2 xx angleACB 180^(@)=2angleACB(therefore angleADB=180^(@))

Knowledge Check

  • The total plane angle subtended by a circle at its centre is

    A
    `pi` rad
    B
    `2pi` rad
    C
    `(2pi)/3` rad
    D
    `pi/2` rad

  • The angle subtended by the diameter at any point on the circles is a//an _______angle.

    A
    acute
    B
    obtuse
    C
    Right
    D
    reflex

  • Which of the following can be the angle subtended by a minor arc at the centre?

    A
    `100^(@)`
    B
    `220^(@)`
    C
    `180^(@)`
    D
    `270^(@)`

    • Similar Questions

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    Consider the following statements 1. the opposite angles of a cyclic quadrilateral are supplementary. 2. Angle subtended by an arc at the centre is doubled the angle subtended by it any point on the remaining part of the circle. Which one of the following is correct in respect of the above statements

    Theorem:-5(i) The degree measure of an arc or angle subtended by an arc at the centre is double the angle subtended by it at any point of alternate segment.(ii) Angle in the same segment of a circle are equal (iii) The angle in the semi circle is right angle.

    Angle subtend by an arc at center is double the angle subtend b it at any point on remaining part OF circle||Angle is a semi circle is a 90 ||Example ||Congurent arc|| Equal arc||Cyclic quadrilateral

    Theorem:-6Any angle subtended by minor arc in the alternate segment is acute and any angle subtended by major arc in the alternate segment is obtuse.

    Any angle subtended by a minor arc in the alternate segment is acute and any angle subtended by a major arc in the alternate segment is obtuse.

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