Question

A chord of length 16 cm is at a distance of 15 cm from the center of the circle find the length of the cord of the same circle which is at a
distance of 8 cm from the centre​

Answer 1

$\huge\mathfrak\blue{Answer:}$

Given:

• We have been given that a chord of length is at a distance of 15 cm from the center of the circle

To Find:

• We have to find the length of chord which is a distance of 8 cm from the center of Circle

Concept Used:

Perpendicular from the center of the circle to the chord, Divides the chord in two equal segments

Solution:

We have been given that

Length of Chord (AB) = 16 cm

Drawing a perpendicular (OD) from the center O to the chord AB

Such that OD = 15 cm

Since OD $\perp$ AB

Perpendicular from the center of the circle to the chord, Divides the chord in two equal segments

$\boxed{\sf{AD = BD = \dfrac{16}{2} = 8 cm }}$

Joining Point O to A [Construction]

$\boxed{\sf{OA = Radius \: of \: Circle }}$

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Using Pythagoras Theorm in △ODA

$\implies \sf{OA^2 = OD^2 + AD^2}$

Substituting the Values in above Equation

$\implies \sf{OA^2 = 15^2 + 8^2}$

$\implies \sf{OA^2 = 225 + 64}$

$\implies \sf{OA^2 = 289}$

Taking Square Root on Both sides

$\implies \sf{OA = \sqrt{289}}$

$\implies \sf{OA = 17 cm}$

$\boxed{\sf{Radius \: of \: Circle = 17 cm}}$

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Let the chord which is at a distance of 8 cm from the center be MN

Drawing a perpendicular (OX) from the center O to the chord MN of circle

Such that OX = 8 cm

Since OX $\perp$ MN

Perpendicular from the center of the circle to the chord, Divides the chord in two equal segments

$\boxed{\sf{MX = NX}}$

Joining Point O to M [Construction]

$\boxed{\sf{OM = Radius = 17 cm }}$

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Using Pythagoras Theorm in △OMX

$\implies \sf{OM^2 = OX^2 + MX^2}$

Substituting the Values

$\implies \sf{17^2 = 8^2 + MX^2}$

$\implies \sf{MX^2 = 289 - 64}$

$\implies \sf{MX^2 = 225}$

$\implies \boxed{\sf{MX = 15 cm }}$

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Length of chord can be determined by

$\implies \sf{Length \: of \: Chord \: = MN}$

$\implies \sf{2 \times MX}$

$\implies \sf{2 \times 15}$

$\implies \sf{30 cm}$

Length of chord which is at a distance of 8 cm from the center is 30 cm

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$\huge\underline{\sf{\red{A}\orange{n}\green{s}\pink{w}\blue{e}\purple{r}}}$

$\large\boxed{\sf{\purple{Length \: of \: chord = 30 cm}}}$

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