Question

the correct answer will get brainliest



the figure, prove that:
11-1
in the adjoining figure, the circles with
centres P, Q and R intersect at points
B.C.D and E as shown. Lines CB and
ED intersect at point M. Lines drawn
from point M touch the circles at
points A and F. Prove that MA= MF.
​

Answer 1

Answer:

construction:-join E to B and C to D

Answer 2

Given:

• There are 3 circles with centres P,Q,R intersect at the points B, C, D and E
• Given that Lines drawn from point M touch the circle at points A and F.

To Find:

• Prove that MA = MF

Solution:

Let us consider the circle with center P

From the figure,it is obvious that MA is tangent and MC is secant.

Then, by tangent secant theorem,

we have,

         $MA^2 = MB \times MC -------------(1)$

Now, Consider the circle with center Q.

From the figure, we can see that the lines MC and ME intersect externally at the point M.

by external division of chord theorem, we have,

           $MC \times MB = ME \times MD$            -----------(2)

Considering the circle with center R, MF is the tangent and ME is the secant.

Then, by tangent secant theorem, we have,

              $MF^2 = MD \times ME$        -----------(3)

Substituting equation (1) and (3) in (2), we get,

                  $MA^2 = MF^2$  

Taking square root on both sides, we have,

           $MA = MF$

Hence proved