algebra-precalculusgeometry
"Two spheres of radii 3 inches and 5 inches rest on a table and touch one another. How far apart are the points at which they touch the table?
I get 3 + $\sqrt{21}$, but the book states the result as 2$\sqrt{15}$.
To solve the problem, I draw a line between the center of the small circle to a point on the radius of the big circle. I then work my way through with the Pythagorean theorem.
Any hints?
Regions P and S are similar, as are regions Q and R. And the area of the right-hand circle is four times the area of the left-hand circle. So $P = 4S$ and $Q = 4R$.
And we are told that $Q+S = 2(P+R)$.
Substituting, $4R+S = 2(4S+R)$. You can take it from there.
Updated to add: Riffing on André's comment (now deleted), this doesn't just work for circles. For instance, it works for Australias too:
Hint:
Apply Descartes theorem:
$$(\frac 1{r_1}+\frac 1 {r_2}+\frac 1{r_3}+\frac 1 {r})^2=2(\frac 1{r_1^2}+\frac 1 {r_2^2}+ \frac 1{r_3^2}+\frac 1{r^2})$$
Where $r_1, r_2, r_3$ are radii of given circles and r is the radii of small(in the middle of circles) and big circle(the one you try to find it's radius).. That is you get a quadratic equation in term of r , solving it gives you the radius.
Best Answer
We are after the length of the thin black line segment, since that's the same as the distance between the points where spheres touch the table. Use the Pythagorean theorem.