Question

$\rm\: Question$

The chords of a circle are parallel to each other and are on either side of the centre of the circle, if the distance between them is 17 cm and the lengths of the chords are 10 cm and 24 cm, find the radius.​

Answer 1

$\large\underline{\sf{Solution-}}$

Let assume that O be the center of circle having radius r cm.

Let further assume that AB and CD are two parallel chords on the either side of the center of circle such that AB = 10 cm and CD = 24 cm

Let OP and OQ be perpendicular drawn from center O on AB and CD respectively.

Now, given that distance between two parallel chords is 8 cm.

It means, PQ = 17 cm

Let assume that OQ = x cm

So, OP = 17 - x cm

Now,

We know, perpendicular drawn from centre bisects the chord.

So, AP = PB = $\rm \: \dfrac{1}{2} AB$ = 5 cm

Also, CQ = QD = $\rm \: \dfrac{1}{2} CD$ = 12 cm

Now, Join OA and OC.

So, OA = OC = r cm

Now, In $\triangle$ OCQ

By using Pythagoras Theorem, we have

$\rm \: {OC}^{2} = {OQ}^{2} + {CQ}^{2}$

$\rm \: {r}^{2} = {x}^{2} + {12}^{2} - - - (1)$

Now, In $\triangle$ OAP

By using Pythagoras Theorem, we have

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

$\rm \: {r}^{2} = {(17 - x)}^{2} + {5}^{2}$

On substituting the value of $\rm \: r^2$ from equation (1), we get

$\rm \: {x}^{2} + {12}^{2} = {17}^{2} + {x}^{2} - 34x + {5}^{2}$

$\rm \: 144= 289 - 34x +25$

$\rm \: 144= 314 - 34x$

$\rm \: 34x= 314 - 144$

$\rm \: 34x= 170$

$\rm\implies \:x = 5 \:$

On substituting x = 5 in equation (1), we get

$\rm \: {r}^{2} = {12}^{2} + {5}^{2}$

$\rm \: {r}^{2} = 144 + 25$

$\rm \: {r}^{2} = 169$

$\rm \: {r}^{2} = {13}^{2}$

$\bf\implies \:r \: = \: 13 \: cm$

Answer 2

GIVEN :-

AB and CD are two parallel chords of circle with AB = 10cm, CD=24cm and opposite side of the center and distance between chords is 17cm

TO FIND :-

radius of circle = ?

SOLUTION :-

Let 'O' is center of circle and 'r' is radius, on x

OM = 17 - x

in ∆OAM

r² = 5² + (17-x)²

in ∆ONC

r² = 12² + x²

• From 1 & 2 5 ^ 2 + (17 - x) ^ 2 = 12 ^ 2 + x ^ 2

289 + x² - 34x - x² = 144 - 25

34x = 289 - 119 34x = 170 = x = 5

substitute the x = 5

r² = 12² + 5² ⇒ r ² = 144 + 25 = r² = 169

r = 13

Hence radius of circle is 13cm.