Question : The distance of a chord OP from the centre of a circle is 6 cm. If the diameter of this circle is 20 cm, then what will be the sum of the radius and length of chord OP?
Option 1: 18 cm
Option 2: 26 cm
Option 3: 16 cm
Option 4: 20 cm
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Correct Answer: 26 cm
Solution :
Let the radius of the circle as \(r\) and the length of the chord OP as \(L\).
Given that the diameter of the circle is 20 cm, the radius is \(r = \frac{20}{2} = 10\) cm.
Use the Pythagorean theorem to find the length of the chord ($L$), where \(d\) is the distance from the centre to the chord.
$L = 2\sqrt{r^2 - d^2}$
Substituting the given values \(r = 10\) cm and \(d = 6\) cm into the formula,
$⇒L = 2\sqrt{(10)^2 - (6)^2} = 2\sqrt{64} = 16 \text{ cm}$
The sum of the radius and the length of the chord OP = $r + L$
= 10 + 16 = 26 cm
Hence, the correct answer is 26 cm.
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