In the given figure, PQ is a transverse common tangent to two circles with centres A and B and of radii 5 cm and 3 cm respectively. If PQ intersects AB at C such that CP = 12 cm, calculate AB.
Circles
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Question

In the given figure, PQ is a transverse common tangent to two circles with centres A and B and of radii 5 cm and 3 cm respectively. If PQ intersects AB at C such that CP = 12 cm, calculate AB.

KnowledgeBoat · Accepted Answer

From figure, AP ⊥ PQ (∵ tangent at a point and radius through the point are perpendicular to each other.) In right angled triangle PAC. ⇒ CA2 = PC2 + AP2 ⇒ CA2 = 122 + 52 ⇒ CA2 = 144 + 25 ⇒ CA2 = 169 ⇒ CA = ⇒ CA = 13 cm. Considering triangles PAC and BCQ, ⇒ ∠APC = ∠BQC = 90° ⇒ ∠PCA = ∠QCB (Vertically opposite angles are equal) △PAC ~ △QBC by AA axiom. Since triangles are similar hence, the ratio of their corresponding sides are equal. ⇒ CB = 7.8 cm. From figure, AB = AC + CB = 13 + 7.8 = 20.8 cm Hence, AB = 20.8 cm.

Mathematics

In the given figure, PQ is a transverse common tangent to two circles with centres A and B and of radii 5 cm and 3 cm respectively. If PQ intersects AB at C such that CP = 12 cm, calculate AB.

Circles

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