One circle has a radius of $5$ and its center at $(0,5)$. A second circle has a radius of $12$ and its center at $(12,0)$. What is the length of a radius of a third circle, which passes through the center of the second circle and both the points of intersection of first two circles.
I have solved this question using the equation for circles but it turns out that there is an easier method to solve this question. Our teacher said that all that is required to solve this is basic $10$-th grade maths. Can anyone tell me how to solve this using $10$-th grade math?
Let $A(0,5)$, $B(12,0)$ and two given circles intersect in the origin $O$ and in the point $C$.
Thus, the needed circle passes trough $A$, $B$, $C$ and $O$,
which is a circle with diameter $AB$ and since $$AB=\sqrt{12^2+5^2}=13,$$ we got the answer: $$\frac{13}{2}.$$
Take the circle with diameter $AB$.
Since $\measuredangle AOB=90^{\circ}$ and $\Delta AOB\cong \Delta ACB$,
we see that this circle passes trough $A$, $B$, $C$ and $O$ and from here he passes trough $O$, $C$ and $B$.