[Guest]

This is my first post, so appologies if it is on the site already.

Two circles, one of radius of 2 inches and one of radius of 1 inch, lie in the same plane spaced 3 inches apart. Two pairs of tangent lines are drawn between them. For each pair of tangents, what is the length of the line segments between the circles?

[Guest]

Each segment is 2*sqrt(2)

Total length = 4*sqrt(2), or 5.657 inches.

That is if "3 inches apart" means 3 inches from center to center.

[Guest]

Each segment is 2*sqrt(2)

Total length = 4*sqrt(2), or 5.657 inches.

That is if "3 inches apart" means 3 inches from center to center.

[Guest]

Actually, I meant that the two circles are separated by a distance of 3".

See Attached.

[plainglazed]

4.5/cos{tan-1(.5/4.5)}?

[Guest]

Actually, I meant that the two circles are separated by a distance of 3".

See Attached.

Oops, read that wrong...then I am getting...

The red ones are each sqrt(35)

The purple ones are each sqrt(27)

Total is 22.224 inches.

[Guest]

To find the length of the crossed tangents (BE on attached diagram):

- draw a line segment (AD) between the two circles that connects their centers

- draw a radius for each circle (AB and DE) that connects to one of the tangent lines; this forms a right triangle

- the two tangent lines cross at a point © that is proportional to the diameters of the two circles; in this case, they cross 2" from the larger circle's edge

- the triangle (ABC on the attached diagram for the larger circle has a known side of 2" and hypoteneuse of 4"

- the other side can be derived as 2 * SQRT(3)

- you can repeat this for the smaller circle, or just recognize that the corresponding triangle for the smaller circle is similar, yielding a length of SQRT

- thus the length of one of the crossed tangent lines is 3*SQRT(3)

For the upper/lower pair, see second diagram

-extend the line of the upper tangent (labeled BE on the solution diagram

- draw a line through the centers of the two circles beginning at the center of the larger circle and extend it until it intersects with the extended upper tangent line (labeled AE)

- draw in a radius AB and CD

- recognizing that ABE and CDE are similar right triangles, you can solve for the unknown length X

- and then using that, solve for BE and DE, and then calculate BD as the difference; the solution is SQRT(37)

• 
  

4.5/cos{tan-1(.5/4.5)}?

I'm not sure. The solution to both sets of lines can be found by regognition of similar triangles. See spoiler.

[plainglazed]

the red lines are √20.5 (same as before but simpler math, thanks pythagorus) and now √18 for the purple lines.

[Guest]

sqrt(148)-sqrt(37) for the red tangent (top to top)

sqrt(12)+sqrt(3) for the purple one (top to bottom)

solved it using the sine theorem in both cases.

[Guest]

sqrt(148)-sqrt(37) for the red tangent (top to top)

sqrt(12)+sqrt(3) for the purple one (top to bottom)

solved it using the sine theorem in both cases.

which is the same as

sqrt(37) and 3*sqrt(3)

[Guest]

which is the same as

sqrt(37) and 3*sqrt(3)

Excellent!!!!!!!

[Guest]

the red lines are √20.5 (same as before but simpler math, thanks pythagorus) and now √18 for the purple lines.

See other post to idamante for confirmation of my answers, which I've had independently vereified. Good luck.

[plainglazed]

My mistake. I used 1 and two as the diameters of the two circles.