SOLUTION: In the diagram to the left, circle with centre O has a radius of 7 cm. Segment AT is tangent to the circle. AO = 25 cm, and AX = XY (this length is labeled m). Find the length of m

Algebra ->  Length-and-distance -> SOLUTION: In the diagram to the left, circle with centre O has a radius of 7 cm. Segment AT is tangent to the circle. AO = 25 cm, and AX = XY (this length is labeled m). Find the length of m      Log On


   

Question 1208908: In the diagram to the left, circle with centre O has a radius of 7 cm. Segment AT is tangent to the circle. AO = 25 cm, and AX = XY (this length is labeled m). Find the length of m.
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Found 2 solutions by ikleyn, mccravyedwin:
Answer by ikleyn(52847) About Me  (Show Source):
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From the diagram, find AO = 18+7 = 25 cm.
Find the length of AT from the right triangle AOT

    AT = sqrt%2825%5E2-7%5E2%29 = sqrt%28576%29 = 24.


Next, consider right triangle XOT.


In this triangle, the leg XT is  AT-m = 24-m cm;

                  the leg OT is 7 cm;

                  the hypotenuse OX is 7+m cm.


Write the Pythagorean equation

    %2824-m%29%5E2 + 7%5E2 = %287%2Bm%29%5E2.


Simplify

    576 - 2*24m + m^2 + 49 = 49 + 2*7*m + m^2,

    576 - 48m = 14m 

    576 = 48m + 14m

    576 = 62m

    m = 576/62 = 288/31 = 9.290 cm  (rounded).


ANSWER.  m = 288%2F31 = 9.290 cm.

Solved.



Answer by mccravyedwin(408) About Me  (Show Source):
You can put this solution on YOUR website!

 

Let z = XT

By the Pythagorean theorem on right triangle ATO
AT%5E2%2BTO%5E2=AO%5E2
%28AX%2BXT%29%5E2%2BTO%5E2=AO%5E2
%28m%2Bz%29%5E2%2B7%5E2=25%5E2
%28m%2Bz%29%5E2%2B49=625
%28m%2Bz%29%5E2=576
m%2Bz=24
z=24-m

By the Pythagorean theorem on right triangle XTO

XT%5E2%2BTO%5E2=OX%5E2=%28XY%2BYO%29%5E2
z%5E2%2B7%5E2=%28m%2B7%29%5E2
%2824-m%29%5E2%2B49=%28m%2B7%29%5E2
576-48m%2Bm%5E2%2B49=m%5E2%2B14m%2B49
576-48m=14m
576=62m
527%2F62=m
288%2F31=m

Exact value of m+=+matrix%281%2C2%2C288%2F31%2C+cm%29 or about 9.29 cm

Edwin