Question 1187404
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In the diagram, circle with centre O has a radius of 5 cm. 
Segment AT is tangent to the circle. AO = 13 cm, and AX = XY (this length is labeled m). Find the length of m.
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            This problem is not difficult.



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First of all, from triangle AOT,  AT = {{{sqrt(13^2-5^2)}}} = {{{sqrt(169-25)}}} = {{{sqrt(144)}}} = 12 cm.


Next, consider triangle XTO.

Its leg XT has the length (12-m)  cm.  It is the tangent segment to the circle O.


Continue XO further to intersection with the circle O.

You will get the long secant of the length  m+5+5 = m + 10 cm.

The outer part of this secant has the length m.


Using well known property of the tangent segment, secant and its outer part, you can write this equation


    {{{XT^2}}} = m*(m+10),  

which is

    {{{(12-m)^2}}} = m*(m+10).


Simplify and find "m"

    144 - 24m + m^2 = m^2 + 10m

    144 = 10m + 24m

    144 = 34m

     m  = {{{144/34}}} = {{{72/17}}}  cm = 4 {{{4/7}}} cm.    <U>ANSWER</U>
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Solved.