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Statement 1: In a circle with center O, chord AB : chord BC = 1 : 3. If angle AOC is 160° ⇒ angle BOC = 120°.

In a circle with center O, chord AB : chord BC = 1 : 3. If angle AOC is 160° ⇒ angle BOC = 120°. Circle, Concise Mathematics Solutions ICSE Class 9.

Statement 2: AB : BC = 1 : 3

⇒ ∠AOC = 3 x ∠AOB

  1. Both the statements are true.

  2. Both the statements are false.

  3. Statement 1 is true, and statement 2 is false.

  4. Statement 1 is false, and statement 2 is true.

Circles

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Answer

We know that,

Ratio of the angles subtended by the chords on the center is equal to the ratio of the length of the chords.

∴∠AOB∠BOC=ABBC⇒∠AOB∠BOC=13⇒3×∠AOB=∠BOC.\therefore \dfrac{\text{∠AOB}}{\text{∠BOC}} = \dfrac{\text{AB}}{\text{BC}}\\[1em] \Rightarrow \dfrac{\text{∠AOB}}{\text{∠BOC}} = \dfrac{1}{3}\\[1em] \Rightarrow 3 \times \text{∠AOB} = \text{∠BOC}.

So, statement 2 is false.

From figure,

⇒ ∠AOC = ∠AOB + ∠BOC

⇒ 160° = ∠AOB + 3∠AOB

⇒ 160° = 4∠AOB

⇒ ∠AOB = 160°4\dfrac{160°}{4}

⇒ ∠AOB = 40° and, ∠BOC = 3 x 40° = 120°.

So, statement 1 is true.

∴ Statement 1 is true, and statement 2 is false.

Hence, option 3 is the correct option.

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