Q5 of 52 Page 10

In Figure 1, AP, AQ, and BC are tangents to the circle. If AB = 5 cm, AC = 6 cm and BC = 4 cm, then calculate the length of AP (in cm). (CBSE 2012)

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Given,


AQ, AP and BC are tangent


AB = 5 cm


AC = 6 cm


BC = 4 cm


Now mark the point S where BC touches the circle.


Let’s BS = Y cm


CS = 4 – Y cm


As AP and AQ are the tangents that touch the exterior points of the circle are equal in length because tangents to the exterior point of a circle are equal in length.


So,


⇒ AP = AQ


⇒ AB + BP = AC + CQ ……….. (i)


⇒ BP = BS and


⇒ CQ = CS ………. (ii)


From (i) and (ii),


AB + BS = AC + CS


5 + y = 6 + (4 – y)


y + y = 6 + 4 – 5


2y = 10 – 5 = 5 cm


y = 2.5 cm


As,


BP = BS = 2.5 cm


CQ = CS = 4 – y = 4 – 2.5 = 1.5 cm


Therefore,


AP = AB + BP


AP = 5 + 2.5 = 7.5 cm


AQ = AC + CQ


AQ = 6 + 1.5 = 7.5 cm


As we can see the length of AP = 7.5 cm

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