Spheres of radius $3$ and $5$ rest on a table and touch each other. How far away are the points that touch the table?

"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?

1 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.

• Very nice solution and sketch! (+1) – user Sep 19 '18 at 9:20
• Now it's clear thanks! I think my problem was that my drawing was not very clear to spot that the hypotenuse is the sum of the two radii. – raltok Sep 19 '18 at 9:23
• @raltok Drawing the right helping lines is an important skill in geometry. Knowing to draw the line connecting the two centers of touching circles and knowing that its length is the sum of the two radii could be a nice thing to take with you. It might not work in every geometry problem with touching circles, but it should be close to the first thing you try. – Arthur Sep 19 '18 at 9:26
• This is why I got Ds in geometry. +1 – Randall Sep 19 '18 at 11:14