Finding the length of a line in a circle when given the lengths of the sides of a triangle inside the circle
First and foremost, sorry if this was a terribly worded question. I'm pretty new to this and haven't quite got the hang of asking specific questions.
In the diagram below, I'd like to find out the length of $AX$ when $AB=6, AC=5,$ and $BC=9$.
At first, I tried to draw a new triangle $AYX$ to solve the problem, but it didn't really help at all.I can't think of any other way of solving the problem, so I'd really appreciate it if someone could help me out.
PS: Sorry for the super-low quality image. I drew it in MS Paint. If anyone here knows a better way to draw math diagrams, I'd love to know.
$\endgroup$ 33 Answers
$\begingroup$Trusting that $A,B,C$ are the centers of the apparent circles and that all apparent points of tangency are in fact points of tangency we define $r_C,r_B$ to be the radii of the circles with centers $C,B$ respectively , and $r=\overline {AX}$ to be the radius of the large circle. We see at once that $$5=r-r_C\quad \quad 6=r-r_B\quad \quad 9=r_C+r_B$$
Adding the first two equations yields $$11=2r-(r_C+r_B)=2r-9\implies \boxed {r=10}$$
$\endgroup$ 2 $\begingroup$We have a system of the equations:
\begin{eqnarray} r_A-r_B &=& 6\\ r_A-r_C &=& 5\\ r_B+r_C &=& 9 \end{eqnarray}
We are interested in $r_A$. If we sum all equations we get $2r_A = 20$, so $r_A =10$.
$\endgroup$ 1 $\begingroup$The diagram is not consistent with the data provided. With the given data we get the radius of circle C to be 5 which locates the point A on the circle C.
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