Question

In figure QR is c common tangent to the given circles, touches externally at point T. The tangent T meet OR at P. If PT = 3.8cm then find the length of QR

(A) 3.8
(B) 7.6
(C) 5.7
(D) 1.9

Answer 1

Hint: We will solve above question by knowing that “Tangents drawn from an external points are equal in length”.
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Here, O is the external point and OA and OB are tangents from this point.
From the property of tangents we can say that OA = OB

Complete step by step solution:
We know that tangents drawn from an external point to a circle are equal.
So,

PT = QP
And it is given that
PT = 3.8cm
So, QP = 3.8cm …..(1)

Here PR and PT are two tangents from same external point P.
So by property of tangents
PR = PT
So, PR = 3.8cm …..(2)
From equation 1 and 2
QP = PR = 3.8cm
But from figure, we can see that line segment QR made of 2 line segments QP and PR.

QR = QP + PR
QR = (3.8 + 3.8)cm$
QR = 7.6cm
Therefore option B i.e. QR = 7.6cm is a correct option.

Note: 1. Before solving problems related to tangent of circle we should remember all the properties and theorems related to the topic.
2. Another thing is to be remember is that "a tangent line never crosses the circle, it just touches the circle."
So if it is crossing the circle we can't use properties of tangents to it.