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Puzzles of the Month + Solutions



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Puzzle
# 129


Steam
locomotive puzzle
The
3 wheels with center O_{1}, O_{2} and O_{3} are
of the same size and tangent to each other. TO_{3}O_{1} 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: ,
general math knowledge.
Category:
Geometry.
Keywords:
tangent, circle, radius.
Related
puzzles:
 A
mathematic shield,
 Achtung
minen.
 La
locomotiva enigmatica
Le 3 ruote di centro O_{1}, O_{2} e O_{3} sono
identiche e tangenti tra loro. TO_{3}O_{1} è un
triangolo retto. Qual è la lunghezza del segmento AB se
i raggi delle ruote valgono ognuna 6 cm?
Parole
chiave: tangente, cerchi, raggio.
Suggerisci un'altra
soluzione Chiudi
 La
loco énigmatique
Les 3 roues de centre O_{1}, O_{2} et O_{3} sont
identiques et tangentes entre elles. TO_{3}O_{1} est
un triangle rectangle. Quelle est la longueur du
segment AB si
le rayon des roues mesure 6 cm?
Mots
clés: tangent, cercle, rayon.
Propose une
autre solution Fermer

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



AB is
a chord of the circle with center O_{2}
MO_{2} 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 TO_{1}O_{3} and MO_{2}O_{3} :
MO_{2} / TO_{1} = O_{2}O_{3} / O_{1}O_{3} =
1/2
MO_{2} =
1/2 x 6 = 3 [cm]
Applying
Pythagorean theorem on the triangle MO_{2}B :
MB = √(6^{2} 
3^{2}) = 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.

The
5 Winners of the Puzzle of the Month are:
Serhat Duran, Turkey  Evis
Hoxa, Albania  Ramnarayan
Panda, India  Marlon
Manto, USA  Shashank
Rathore, India
Congratulations!



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:
(n^{4}  6n^{3} + 23n^{2} 
18n + 24)/24
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)
Discuss
the problem on our FaceBook
page!

© 2012 G.
Sarcone, www.archimedeslab.org
You can reuse 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.

Puzzle
of the Month by Gianni
A. Sarcone is licensed under a Creative
Commons AttributionNonCommercialNoDerivs 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! 




