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Previous Puzzles of the Month + Solutions

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Send to a Friend Puzzle # 129

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Steam locomotive puzzle
The 3 wheels with center O1, O2 and O3 are of the same size and tangent to each other. TO3O1 is a right triangle. If the radius of the wheels is 6 cm long what is then the length of the segment AB?

Difficulty level: bulbbulb, general math knowledge.
Category: Geometry.
Keywords: tangent, circle, radius.
Related puzzles:
- A mathematic shield,
- Achtung minen.

Source of the puzzle:
© G. Sarcone.
You cannot reproduce any part of this page without prior written permission.
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Solution to puzzle 129

AB is a chord of the circle with center O2
MO2 is perpendicular to AB. In a circle, a radius perpendicular to a chord bisects the chord (and the arc). Thus, AM = MB = AB/2

From the similarity of triangles TO1O3 and MO2O3 :
MO2 / TO1 = O2O3 / O1O3 = 1/2
MO2 = 1/2 x 6 = 3 [cm]

Applying Pythagorean theorem on the triangle MO2B :
MB = √(62 - 32) = 3√3
AB = 2 x MB = 6√3 [cm]

Geometric definitions
Tangent: In classical geometry, the tangent line (or simply the tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point.
Radius: In classical geometry, the radius of a circle or sphere is any line segment from its center to its perimeter.
Chord: In classical geometry, a chord is a geometric line segment whose endpoints both lie on the circumference of the circle.

cup winnerThe 5 Winners of the Puzzle of the Month are:
Serhat Duran, Turkey india flag - Evis Hoxa, Albania Albanian flag - Ramnarayan Panda, India india flag - Marlon Manto, USA USA flag - Shashank Rathore, India india flag


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Beyond the challenge

A problem, sometimes known as Moser's circle problem, asks to determine the number of regions into which a circle is subdivided if n points on its circumference are joined by chords. The answer is:
  (n4 - 6n3 + 23n2 - 18n + 24)/24

example 1

The first values are then 1, 2, 4, 8, 16, 31, 57, 99, 163, 256, 386, 562, ... (This is often given as an example of what happens if you attempt to guess a sequence from the first few terms since this sequence starts with 1, 2, 4, 8, 16, but the next term is 31 and not 32 as expected)

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© 2012 G. Sarcone,
You can re-use content from Archimedes’ Lab on the ONLY condition that you provide credit to the authors (© G. Sarcone and/or M.-J. Waeber) and a link back to our site. You CANNOT reproduce the content of this page for commercial purposes.

Creative Commons License
Puzzle of the Month by Gianni A. Sarcone is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
You're encouraged to expand and/or improve this article. Send your comments, feedback or suggestions to Gianni A. Sarcone. Thanks!
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Puzzle 128:
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Puzzle 127: Square vs Annulus (2011)
Puzzle 126: A troublesome sequence (2011)
Puzzle 125: A mathematical shield (2011)
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Oct-Nov 09:
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May-June 09:
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Jan 07:
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