SOLUTION: https://imgur.com/hrQkDYl (copy and paste this link into the search bar to view the diagram). In the diagram, circle with centre O has a radius of 5 cm. Segment AT is tangent to th

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Question 1187404: https://imgur.com/hrQkDYl (copy and paste this link into the search bar to view the diagram). 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.
Answer by ikleyn(52814) About Me  (Show Source):
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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. 


First of all, from triangle AOT,  AT = sqrt%2813%5E2-5%5E2%29 = sqrt%28169-25%29 = sqrt%28144%29 = 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%5E2 = m*(m+10),  

which is

    %2812-m%29%5E2 = m*(m+10).


Simplify and find "m"

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

    144 = 10m + 24m

    144 = 34m

     m  = 144%2F34 = 72%2F17  cm = 4 4%2F7 cm.    ANSWER

Solved.