Question

Two concentric circles are of radii 5 cm and 3 cm .Find the length of the chord of the large circle which touches the smaller circle?

Answer 1

Answer:

$\boxed{\sf \: The\:length\:of\:chord\:is\:8\:cm \: }$

Step-by-step explanation:

Let assume that C(O, r) and C(O, R) be two concentric circles with centre O and such that R = 5 cm and r = 3 cm.

Let further assume that AB be a chord of larger circle touches the small circle at C.

Construction :- Join OC

We know, radius and tangent are perpendicular to each other.

So, OC is perpendicular to AB.

Now, as OC is perpendicular to AB and we know, perpendicular drawn from centre bisects the chord.

So, AC = CB

Now, In right-angle triangle OAC

By using Pythagoras Theorem, we have

$\sf \: {OA}^{2} = {OC}^{2} + {AC}^{2}$

$\sf \: {R}^{2} = {r}^{2} + {AC}^{2}$

$\sf \: {5}^{2} = {3}^{2} + {AC}^{2}$

$\sf \: 25 = 9 + {AC}^{2}$

$\sf \: {AC}^{2} = 25 - 9$

$\sf \: {AC}^{2} = 16$

$\bf\implies \:AC = 4\: cm$

Now,

$\bf\implies \:AB = 2AC = 8\: cm$

Hence,

$\implies\sf \: \boxed{\sf \: The\:length\:of\:chord\:is\:8\:cm \: }$

Answer 2

Answer:

$\boxed{\sf {Length\:of\:the\:chord\:=\:8 cm}}$

Step-by-step explanation:

Given: Two concentric circles have radius 5 cm and 3 cm

To find: The length of chord of circle which touches the smaller circle.

OA = $\sf\:3$cm (radius of smaller circle)

OP = $\sf\:5$cm (radius of bigger circle)

Let O be the center of two concentric circles.

Let PQ be the chord of the larger circle which touches the smaller circle at point A. Here, PQ is tangent to the smaller circle.

OA is perpendicular to PQ (OA is the radius of the circle)

Now, we have to find the length of chord of circle which touches the smaller circle by Pythagoras theorem:

In Δ ΟΑΡ,

OA² + AP² = OP² [By Pythagoras theorem]

3² + AP² = 5³

9 + AP² = 25

AP² = 25 - 9

AP² = 16

AP = $\sf \sqrt{16}$

AP = 4 cm

Since OA is perpendicular to PQ,

AP= AQ (Perpendicular from the center of the circle bisects the chord)

PQ = 2AP

PQ = 2 × 4

PQ = 8cm

Hence, the length of the chord of the larger circle is 8 cm.

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Concentric circles:

Circles having the same centre but different radii are called Concentric circles.

Chord :

A line segment joining any two points on a circle is called a chord of the circle.