Question

two concentric circles of radii 5cm and 3cm ​

Answer 1

$\: \: \: \: \: \: \: \: \star\bf\: \: \: {Required\: figure}$

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Given

• Radius of smaller circle = 3cm
• Radius of larger circle = 5cm

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To find

• Length of chord of the larger circle.

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Solution

• According to the figure, we need to find the length of PQ.

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★ As we can see that ∆OAP is a right angled triangle.

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• Therefore, in triangle OAP.

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$\: \: \: \: \: \: \: \: \: \:\underline{\sf{\red{By\: using\: Pythagoras\: theorem}}}$

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$\tt:\implies\: \: \: \: \: \: \: \: {OP^2 = OA^2 + AP^2}$

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$\tt:\implies\: \: \: \: \: \: \: \: {(5)^2 = (3)^2 + AP^2}$

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$\tt:\implies\: \: \: \: \: \: \: \: {25 = 9 + AP^2}$

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$\tt:\implies\: \: \: \: \: \: \: \: {AP^2 = 25 - 9}$

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$\tt:\implies\: \: \: \: \: \: \: \: {AP^2 = 16}$

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$\tt:\implies\: \: \: \: \: \: \: \: {AP = \sqrt{16}}$

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$\tt:\implies\: \: \: \: \: \: \: \: {AP = 4\: cm}$

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• As we can see in the figure, that OA is the perpendicular bisector of PQ.

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$\star{\boxed{\sf{\orange{PQ = 2AP}}}}$

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$\tt\longrightarrow{PQ = 2 \times 4}$

$\tt\longrightarrow{PQ = 8\: cm}$

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Therefore,

• The length of chord of the larger circle is 8cm.

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Hence,

• Option (c) is the correct option.

Answer 2

★Given that,two concentric circles are of radii 5cm and 3cm.

To Find:- length of the chord of the larger circle, (in cm), which touches the smaller circle is.?

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solution:

now,let the radius of the bigger circle be R and the radius of the smaller circle be r

as it is given that R= 5cm = OB and r = 3cm= OD

and AB is the chord whose length is to be found.

AB=BD and OD ⟂ AB

therefore,triangle OBD is right angled.

where, OD²+BD²=OB²

$⟼{3}^{2} + {bd}^{2} = {5}^{2} \\ ⟼9 + {bd}^{2} = 25cm \\ ⟼ {bd}^{2} = 25 - 9 cm\\ ⟼ {bd}^{2} = 16cm \\ ⟼ \sqrt{bd} = 4cm \\ ⟼db = \mathbb{\boxed{\purple{4cm}}\star}$

Since, AD= BD=4cm

therefore ,AB=AD+BD

⇒AB= 4+4cm

⇒AB=$\mathbb{\boxed{\purple{8cm}}}$

hence option c 8cm is correct.!

hope this helps.!!