SOLUTION: A bicycle chain fits tightly around two gears.What is te distance between the centers of the gears if the radii of the bigger and smaller gears are 9.3 inches and 2.4 inches,respec

Question 1061647: A bicycle chain fits tightly around two gears.What is te distance between the centers of the gears if the radii of the bigger and smaller gears are 9.3 inches and 2.4 inches,respectively,and the portion of the chain tangent to the two gears is 26.5 inches long?
Found 3 solutions by josgarithmetic, ikleyn, MathTherapy:
Answer by josgarithmetic(39620)   (Show Source): You can put this solution on YOUR website!
This is not the whole solution answer, but a way to begin.

The TANGENT line. Draw first just the tangent line. You know you will need two circles, so now draw each of them. Show one circle, and some distance away, show the other circle. Distance between the two tangency points is 26.5 as given. The center of each circle to its tancency point is the radius of the circle. The smaller of these is 2.4 units and the larger of these is 9.3 units. Now you have THREE known side lengths of a trapezoid.

Note also that each of the perpendicular radii make 90 degree angle to the tangent line.
Put all this onto a cartesian coordinate system! Make the tangent line be contained in the x-axis. If you put the two perpendicular radii extending toward the positive y-direction; and if you align the left-most circle's radius on the y-axis, then your centers of your circles will have two identifiable points. What is the distance between these two points, forming the unknown length of the trapezoid? Use the Distance Formula.

Can you follow that description?

Let this be a way to form the figure on Cartesian system:
(0,0) tangency point for small circle, and center is at (0,2.4).
(26.5,0) tangency point for big circle, and center is at (26.5,9.3).
If put these into the Distance Formula, this expression is the distance between the centers of the circles.
Answer by ikleyn(52802)   (Show Source): You can put this solution on YOUR website!
.

Answer. d = = = 27.38 in     (approximately).

See the figure in the lesson
    - HOW TO construct a common exterior tangent line to two circles
in this site.

The figures and the formula explain everything without words.

The Pythagorean theorem is used, as well as the property

        "If a tangent line is drawn to the circle, then the radius of the circle to the tangent point is perpendicular to the tangent line".

O ! One more property is used: The two perpendiculars to a straight line are parallel.

Shortly, if you know basic elementary facts of Geometry, you will be able to reconstruct the solution (the arguments) in your mind.

Also, you have this free of charge online textbook on Geometry
    GEOMETRY - YOUR ONLINE TEXTBOOK
in this site.

The referred lesson is the part of this textbook under the topic
"Properties of circles, inscribed angles, chords, secants and tangents".

Answer by MathTherapy(10552)   (Show Source): You can put this solution on YOUR website!

A bicycle chain fits tightly around two gears.What is te distance between the centers of the gears if the radii of the bigger and smaller gears are 9.3 inches and 2.4 inches,respectively,and the portion of the chain tangent to the two gears is 26.5 inches long?

I was about to do this problem and remembered that it'd been done previously.
I don't know if you or someone else had submitted it, but this is a duplicate of problems: Circles/997071, and Circles/996970.
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