Question

Two concentric circles are of radii 5cm and 3cm. Find the length of the chord of the larger circle which touches the small circle.


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Answer 1

$\LARGE\mathfrak\pink{answer}$

• Let O be the centre of the concentric circles. And AB be a chord of the larger circle touching the smaller circle P.

Then,

AP = PB and OP ⊥ AB

Applying Pythagoras Theorem in △ OPA, we have

OA² = OP² + AP²

$\mapsto$ 25 = 9 + AP²

$\mapsto$ AP² = 25 - 9

$\mapsto$ AP² = 16 $\rightarrow$ 4 cm

$\mapsto$ AP = 2AP = 8cm.

$\large\sf\underbrace\orange{∴\:chord\:AB\:measures\:8cm.}$

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# Refer the diagram also.

# Be Brainly :)

Answer 2

★ Concept :-

Here the concept of Tangents and Pythagoras Theorem have been used. We see that we are given two concentric circles where the chord of bigger circle touches the smaller circle externally at a point. This means that the chord is like a tangent to the smaller circle. On knowing this, we can apply different propeties of Tangents and then find relationship between different given lengths. After that we shall apply Pythagoras Theorem here and thus find the length of the chord.

Let's do it !!

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★ Formula Used :-

$\;\;\boxed{\sf{\pink{(Hypotenuse)^{2}\;=\;(Base)^{2}\;+\;(Height)^{2}}}}$

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★ Solution :-

Given,

» Radius of bigger circle = 5 cm

» Radius of smaller circle = 3 cm

» There lies a chord of bigger circle touching the smaller circle.

Let's now understand the figure of this question.

(Refer to the attachment).

From figure, we can get that

>> O is the centre of the concentric circles.

>> AB is the chord of the bigger circle.

>> AB touches smaller circle at point C.

>> Length of OA = radius of bigger circle = 5 cm

>> Length of OC = radius of smaller circle = 3 cm

• Theorem I : Radius of the circle is perpendicular to the tangent to the circle and bisects the tangent where it meets the tangent.

Here the radius of smaller circle meéts it's tangent AB at the point C . This means,

>> AC = BC (By Theorem I)

• Let AC be x cm

Then,

>> AC = BC = x

>> AB = AC + BC = x + x = 2x

Clearly radius is perpendicular to the tangent. Which means that, OC is perpendicular to AB. This means,

• <OCA = <OCV = 90°

So ∆AOC is a right - angled triangle.

This means here we can use Pythagoras Theorem here.

By mathematical formulation of Pythagoras Theorem, we know that

$\;\;\tt{\rightarrow\;\;(Hypotenuse)^{2}\;=\;(Base)^{2}\;+\;(Height)^{2}}$

• Here Hypotenuse = AO = 5 cm

• Here Height = OC = 3 cm

• Here Base = AC = x cm

By applying values, we get

$\;\;\tt{\rightarrow\;\;(5)^{2}\;=\;(x)^{2}\;+\;(3)^{2}}$

$\;\;\tt{\rightarrow\;\;25\;=\;x^{2}\;+\;9}$

$\;\;\tt{\rightarrow\;\;x^{2}\;=\;25\;-\;9}$

$\;\;\tt{\rightarrow\;\;x^{2}\;=\;16}$

$\;\;\tt{\rightarrow\;\;x\;=\;\sqrt{16}}$

$\;\;\tt{\rightarrow\;\;x\;=\;\pm\:4}$

• Here we will neglect the negative value of x since, x is a distance and distance is always positive.

$\;\;\bf{\rightarrow\;\;\red{x\;=\;4\;\;cm}}$

Now the length of chord is given as,

>>> AB = x + x = 2x = 2(4) = 8 cm

This is the answer.

$\;\;\underline{\boxed{\tt{Length\;\:of\;\:chord\;\:=\;\bf{\purple{8\;\:cm}}}}}$

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★ More to know :-

• Tangent : It is a line from an external that touches a circle externally at any one point.

• Circle : It is a collection of infinite points on a plane joined by a line which has no ending and starting point. Also the distance from circle to any point on the line is same always.

• Triangle : It is a closed figure of three sides where the sum of all three angles equals to 180°.

• Right - Angled Triangle : It is a triangle where any one angle equals to 90° and other angles are acute angle.

• Concentric Circles : These are the circles whose radius varies but the centre is the same point on same plane.