Help with a tangent circle/line problem, please!

[Guest]

Hello all, thanks in advance for reading my post!

Okay, the problem is "There are five circles tangent to each other, and all five circles are also tangent to two lines (one on top, one on bottom...) If the largest circle has radius of 18, and the smallest has a radius of 8, what is the radius of the middle circle?" Here's my best drawing of the picture in the problem:

OOOOO

And then the line on top (which I can't figure out how to draw here...) is simply tangent to each circle, thus is at an angle, while the bottom one is just straight, as shown.

My first thoughts for how to solve this problem, were to use radii of the smallest an largest circles, and simply average them? But then I thought that each circle might not increase by an equal interval... and then again, they might... And I'm just really stuck, and can't remember seeing a problem like this before! Thanks in advance for all your help, for it's most appreciated!

 

[tkhunny]

Have you considered drawing a line through all the circles' centers? Can it be done with a single line?

Note: "straight"?? I think you mean horizontal.

 

[Guest]

Have you considered drawing a line through all the circles' centers? Can it be done with a single line?

Note: "straight"?? I think you mean horizontal.

Ha, yes, I do mean horizontal, I apologize! And, I did think about drawing a line through the centers, but isn't that the same as the tangential line on the top of each circle? They'd be the same length, and at the same angle...

 

[chapdawg147]

I am not really sure how to post pictures so you will try and explain with words.

I am pretty sure I am doing this correct, mostly becuase I came out with a whole number as the answer. The size of each circle is defined by the tangent points with respect to each line and the tangent with respect to the previous circle.

This means that the size of each circle is dependent on the previous circle.

Because the circle size is restricted by the top and bottom lines, the size increase for each circle should be the same (I am pretty sure this is a correct assumption, if this is not I have wasted you time).

The equation for the second circle's radius is

circle1*x=circle2

the equation for circle 3 is

circle2*x=circle3

but circle2=circle1*x

so by subsitution

circle3=circle1*x*x

If you contiue this trend you should be able to make the equation in terms of circle1 and circle5 who have known radii. solve for x and then you should be able to solve for circle3, the middle circle. The answer I got was 12.

 

[stapel]

Here's my best drawing of the picture in the problem...

So you mean something like this...?

                                                  _____
                                              ,'         ',
                             ,----,         /               \
                           '        '     /                   \
                __       /            \  |                     |
             ,'    ',   '              ' |                     |
      ___   /        \ |                ||                     |
 __  /   \ |          | '              '  \                   /
/  \|     | \        /   \            /     \               /
\__/ \___/   ', __ ,'      ' ,____, '         ',  ______ ,'
----------------------------------------------------------------

...but with an additional "slanty" line tangent to all the circles across the "top"?

...can't remember seeing a problem like this before!

You won't always be given exercises exactly like the worked examples. By this age, you're expected to be able to use what you've learned to puzzle out new things for yourself. Just FYI....

You've certainly seen tangents, right angles, right triangles, and ratios, and probably systems of equations (back in algebra). So use that.

There are probably many ways to solve this, but one might be to follow the previous suggestion: Draw the line through the centers of the circles. Then draw the perpendicular segments from the centers to the tangents. You know that the two tangents must intersect. Draw this, and label this length as "x". Label the segments as "8" and "18" for the circles on the ends, and "a", "b", and "c" for the circles in between. (You are looking for the value of "b".)

Since the right triangles formed by the tangents, the line through the centers, and the segments are, of necessity, similar, you can form the ratios relating the sides of the similar figures. For instance, the first and second triangles (from the left) give you "x/8 = (x + 8 + a)/a".

See what you can do about solving the system of equations.

Eliz.