Question

PQ and PR are two equal chords of a circle with centre O. If
PQ = 5 cm and the radius of the circle is 6 cm, find the length of
the chord QR.

plz solve ​

Answer 1

Correct Question :

In the figure given above , PQ and PR are two equal chords of a circle with centre O.

If PQ = 5 cm and the radius of the circle is 6 cm, find the length of

the chord QR.

Solution :

Here , join the line segments OQ and OR .

Let line segment OP intersect the segment QR at point S .

We can clearly observe that ∆ OQS and ∆ORS, ∆ PQS and ∆ PRS are right angled congruent triangles.

Applying the Pythagoras theorem here in ∆ OQS ;

=> OS² + QS² = OQ²

And OQ = OR ( radii of the circle )

Suppose that the length QR is a units .

QS = a/2

OS² + a²/4 = 36

OS² = 36 - a²/4

OS = √{ 36 - a²/4 }

PS = 6 - √{ 36 - a²/4 }

Applying the Pythagoras theorem here as well ;

{ 6 - √{ 36 - a²/4 } }² + a²/4 = 25

let a²/4 = k

=> ( 6 - √{ 36 - k } )² + k = 25

=> 36 + 36 - k - 12√ 36 - k + k = 25

=> 72 - 12√ 36 - k = 25

=> 12 √ 36 - k = 47

=> √ 36 - k = 47/12

=> 36 - k = ( 47/12)²

=> k = 36 - ( 47/12)²

=> a²/4 = 36 - ( 47/12)²

=> a² = 4 [ 36 - ( 47/12)² ]

=> a =2 √{ 36 - ( 47/12)² }

We get the value of a as 9 cm approx.

This is the required answer.

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