Answer :

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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