is a chord of circle with centre . At , a tangent is drawn such that its length is 24 . The distance of from the centre is 26 . If the chord is 16 , find its distance from the centre.

We need to find the distance of chord from the centre if the chord is 16 . On joining and drawing ⊥, the figure can be drawn as shown belowFrom ∆, on using Pythagoras theorem, we get2 = 2 + 2On substituting the values, we get⇒ 262 = 2 + 242⇒ 676 = 2 + 576⇒ 2 = 100⇒ = 10 It is known that the perpendicular drawn from the centre to the chord, divides the chord into two equal parts. Hence, = 8 Similarly, from ∆, on using Pythagoras theorem, we get2 = 2 + 2On substituting the values, we get⇒ 102 = 2 + 82⇒ 100 = 2 + 64⇒ 2 = 36⇒ = 6 Hence, the distance of the chord from the centre is 6 .

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𝐴𝐵 is a chord of circle with centre 𝑂. At 𝐵, a tangent 𝑃𝐵 is drawn such that its length is 24 𝑐𝑚. The distance of 𝑃 from the centre is 26 𝑐𝑚. If the chord 𝐴𝐵 is 16 𝑐𝑚, find its distance from the centre.

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Detailed Solution

We need to find the distance of chord from the centre if the chord 𝐴𝐵 is 16 𝑐𝑚. On joining 𝑂𝐵 and drawing 𝑂𝐶 $⊥$ 𝐴𝐵, the figure can be drawn as shown below

From ∆𝑂𝐵𝑃, on using Pythagoras theorem, we get
𝑂𝑃² = 𝑂𝐵² + 𝐵𝑃²
On substituting the values, we get
⇒ 26² = 𝑂𝐵² + 24²
⇒ 676 = 𝑂𝐵² + 576
⇒ 𝑂𝐵² = 100
⇒ 𝑂𝐵 = 10 𝑐𝑚
It is known that the perpendicular drawn from the centre to the chord, divides the chord into two equal parts. Hence, 𝐵𝐶 = 8 𝑐𝑚
Similarly, from ∆𝑂𝐵𝐶, on using Pythagoras theorem, we get
𝑂𝐵² = 𝑂𝐶² + 𝐵𝐶²
On substituting the values, we get
⇒ 10² = 𝑂𝐶² + 8²
⇒ 100 = 𝑂𝐶² + 64
⇒ 𝑂𝐶² = 36
⇒ 𝑂𝐶 = 6 𝑐𝑚
Hence, the distance of the chord from the centre is 6 𝑐𝑚.