Question : A chord of length 40 cm is at a distance of 15 cm from the centre. What is the length of the chord of the same circle which is at a distance of 20 cm from the centre of the circle?
Option 1: 15 cm
Option 2: 25 cm
Option 3: 45 cm
Option 4: 30 cm
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Correct Answer: 30 cm
Solution :
The length of the chord can be calculated using the Pythagorean theorem. If we consider the radius of the circle as $r$, the distance of the chord from the centre as $d$, and the length of the chord as $l$.
$ r^2 = \left(\frac{l}{2}\right)^2 + d^2 $
From the given problem, we know that a chord of length 40 cm is at a distance of 15 cm from the centre.
$ ⇒r^2 = \left(\frac{40}{2}\right)^2 + 15^2 = 20^2 + 15^2 = 625 $
$⇒r= 25 \text{ cm}$
Let the length of the chord which is at a distance of 20 cm from the centre as $l' $.
$ ⇒25^2 = \left(\frac{l'}{2}\right)^2 + 20^2 $
$⇒ l' = 2 \sqrt{25^2 - 20^2} = 2 \sqrt{625 - 400} = 2 \sqrt{225} = 2 \times 15 = 30 \text{ cm} $
Hence, the correct answer is 30 cm.
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