Question

In the given figure AB and CD are two common P. tangents of two circles with cen- tres P and Q. These circles tou-ch each other at M. If the common tangent at M meets AB and CD at X and Y respectively, prove that XY =1/ 2 (AB + CD) ​

Answer 1

Step-by-step explanation:

Given that Ab and CD are two common tangents of two circles with centres P and Q. These circles touches each other at point M. The common tangent at M meets Ab and CD at X and Y respectively.

We need to prove that XY = 1/2 (AB + CD).

We know that tangents drawn from an exterior points are equal. So,

For tangent AB:

→ XM = XB (circle having centre Q)

→ AX = XM (circle having centre P)

From above we can say that, XM = XB = AX.

Similarly, for tangent CD:

→ MY = YD (circle having centre Q)

→ MY = YC (circle having centre P)

From above we can say that, MY = YD = YC.

Also, AX = 1/2 AB. And we already discussed above that XM = XB = AX. Therefore, XM = 1/2 AB and similarly MY = 1/2 CD.

Point M divides the line XY into XM and MY.

→ XY = XM + MY

→ XY = 1/2 × AB + 1/2 × CD

→ XY = 1/2 (AB + CD)

Hence, proved.