Question : If a chord of length 16 cm is at a distance of 15 cm from the centre of the circle, then the length of the chord of the same circle which is at a distance of 8 cm from the centre is equal to:
Option 1: 10 cm
Option 2: 20 cm
Option 3: 30 cm
Option 4: 40 cm
Correct Answer: 30 cm
Solution : Given: O is the centre of a circle. AB = 16 cm is a chord of the given circle and AB is at a distance of 15 cm from O. Another chord CD is at a distance of 8 cm from O. We join OA and draw a perpendicular ON to AB. So, $\angle$ONA = 90° So, OA is the radius of the circle. Now AB = 2AN, since the perpendicular, drawn from the centre of a circle to the chord, bisects the chord. So, AN = $\frac{16}{2}=8$ cm Given, ON = 15 cm By using Pythagoras's theorem, we get, OA = $\sqrt{AN^2+ON^2}=\sqrt{8^2+15^2}=17$ cm Now, in ΔOMC, OM = 8 cm and OC = OA = 17 cm (radii of the same circle). Also, $\angle$OMC= 90°, since OM is the perpendicular distance of CD from O. Using Pythagoras theorem, we get, CM = $\sqrt{OC^2-OM^2}=\sqrt{17^2-8^2}=15$ cm. M bisects CD in half. So, CD = 2CM = 2 × 15 = 30 cm So, the length of the chord is 30 cm. Hence, the correct answer is 30 cm.
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