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Two Circle with Centres a and B, and Radii 5 Cm and 3 Cm, Touch Each Other Internally. If the Perpendicular Bisector of the Segment Ab Meets the Bigger Circle in P and Q; Find the Length of Pq. - ICSE Class 10 - Mathematics

ConceptChord Properties - the Perpendicular to a Chord from the Center Bisects the Chord (Without Proof)

Question

Two circle with centres A and B, and radii 5 cm and 3 cm, touch each other internally. If the perpendicular bisector of the segment AB meets the bigger circle in P and Q; find the length of
PQ.

Solution If two circles touch internally, then distance between their centres is equal to the difference of their radii. So, AB = (5 − 3) cm = 2 cm.
Also, the common chord PQ is the perpendicular bisector of AB. Therefore, AC = CB =1/2 AB
= 1 cm
In right ΔACP, we haveAP^2 = AC^2 + CP^2

⇒ 5^2 = 1^2 + CP^2

⇒ CP^2 = 25 - 1 = 24

⇒ cp = sqrt(24)= 2 sqrt(6)  cm

Now , PQ = 2 CP

= 2xx 2 sqrt(6)  cm

= 4 sqrt (6 )  cm

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Solution Two Circle with Centres a and B, and Radii 5 Cm and 3 Cm, Touch Each Other Internally. If the Perpendicular Bisector of the Segment Ab Meets the Bigger Circle in P and Q; Find the Length of Pq. Concept: Chord Properties - the Perpendicular to a Chord from the Center Bisects the Chord (Without Proof).
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