Question : A chord of length 48 cm is at a distance of 7 cm from the centre of the circle. What is the length of the chord of the same circle which is at a distance of 15 cm from the centre of the circle?
Option 1: 40 cm
Option 2: 45 cm
Option 3: 35 cm
Option 4: 42 cm
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Correct Answer: 40 cm
Solution :
The perpendicular from the centre of a circle to a chord bisects the chord.
By Pythagoras theorem:
Hypotenuse
2
= Base
2
+ Perpendicular
2
MP = PN = $\frac{\text{MN}}{2}$ = $\frac{48}{2}$ = 24 cm
Distance from the centre as $d_{1}$ = 15 cm
From the Pythagorean theorem, the radius $r$ of the circle is:
$r = \sqrt{a^{2}+d_{1}^{2}}$
⇒ $r = \sqrt{24^{2}+7^{2}}$
⇒ $r = \sqrt{576+49}$
⇒ $r = \sqrt{625}$
⇒ $r= 25$ cm
For the unknown chord, let's denote half of its length as $b$ and the known distance from the centre as $d_{2}$ = 15 cm
Again using the Pythagorean theorem, we find $\frac{x}{2}$ as
$\frac{x}{2} = \sqrt{r^{2}-d_{2}^2}$
⇒ $\frac{x}{2} = \sqrt{25^{2}-15^2}$
⇒ $\frac{x}{2} = \sqrt{400}$
⇒ $\frac{x}{2} = 20$ cm
So, the full length of the unknown chord is $x$ = 2 × 20 cm = 40 cm
Hence, the correct answer is 40 cm.
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