Question

In Fig. 10.87, two equal circles touch each other at T, if QP = 4.5 cm, then QR =
A. 9 cm
B. 18 cm
C. 15 cm
D. 13.5 cm

Answer 1

To calculate the length of QR, we have to keep in mind that the two circles are having the same radius.

Explanation:

So here ,

Given: - The two circles have same radius [AS they are equal].

‘T’ is the point where the two  circles meet.

QP = 4.5 cm.

We have to find the length of QR.

Now, when looking at the figure,  

For circle 1.  

P is a point that is outside the circle and touches the circle at points Q and T.

So, PQ and PT are the tangents to the circle.

Thus PQ = PT. --------------(1)

For Circle 2.

P is a point that is again outside the circle and touches the circle at R and  at T.

So, PR and PT are the two tangents to the circle.

Thus, PR = PT. -----------------(2)

But, by (1) and (2), we have, QP = PT and PR = PT

So, QP = PR. ----------------(3)

And QR = QP + PR

So, by (3), we can also say that QR = QP + QP (as PR = QP).

Or, QR = 2QP.

Or, QR = 2 x 4.5  (as QP = 4.5 cm).

Or, QR = 9 cm.

So the correct answer is Option A.