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Send to a Friend Puzzle # 128
puzzle no. 128

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Xmas tree geometric puzzle
Find the value of h.

Difficulty level: bulb, general math knowledge.
Category: Geometry.
Keywords: trapezium, trapezoid, base.
Related puzzles:
- Prof Gibbus' angle,
- 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 128

This is not a one-solution puzzle, as according to the information given there is a range of possible solutions.
a) The shape delimited on the Xmas tree forms a trapezium/trapezoid.
b) The opposite angles of the lower base add together and equal 90 degrees. Then, if we extend the legs of the trapezium/trapezoid they will meet at an angle of 90 degrees. All the possible 90-degree angles are inscribed in a semicircle (Thales’ theorem, see diagram above). In consequence, the height h is maximum when any of the angles of the lower base equals 45 degrees.
c) In this case, hmax = (54 – 21)/2 = 16.5 [cm]

Thus, the value x of h is: 16.5 [cm] > x > 0 [cm]

Geometric terms
Trapezoid: N. Amer. a quadrilateral with only one pair of parallel sides.
Trapezium: Brit. a quadrilateral with only one pair of parallel sides.
Base: one of the parallel sides of the trapezium/trapezoid. Every trapezium/trapezoid has two bases.
Leg: one of the sides of the trapezium/trapezoid that are not parallel. Every trapezium/trapezoid has two opposite legs.

cup winnerThe Winner of the Puzzle of the Month is:
Gordon Steadman, Canada canada flag


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Beyond the challenge (#128bis)

A small change has been made to the statement of the above-mentioned math problem:
ABCD is a trapezium/trapezoid
α ≠ 45°
AB = 21 [cm]
DC = 54 [cm]
AJ = JB and DK = KC
Now, find the value x of JK (this is a one-solution puzzle! See diagram below).

example 1

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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.

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 127: Square vs Annulus (2011)
Puzzle 126: A troublesome sequence (2011)
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Aug-Sept 2006
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Apr-May 06:
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