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Acute Geometry Problem

I saw a plea for help on Twitter from Astronomer Bryan Gaensler who is stuck with his son’s homework.

So please give him a hand by solving this to find a, b and c.

Your time starts now.

This entry was posted on December 3, 2018 at 12:37 pm and is filed under Cute Problems, mathematics. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

15 Responses to “Acute Geometry Problem”

  1. David's avatar
    David Says:
    December 3, 2018 at 2:08 pm

    a=1, b=15, c=22.5

    Reply
  2. Anton Garrett's avatar
    Anton Garrett Says:
    December 3, 2018 at 2:24 pm

    To see that there is not a unique answer, imagine a rod of length 7.5 units, attached by a hinge to a rod of length 2 units, which is itself attached at its other end by a hinge to a rod of length 3 units. These hinges are such that the three rods are constrained to a plane. Suppose that the first-mentioned hinge is at the apex of the diagram, and the plane of the rods is the plane of the paper without loss of generality. Then the free end of the 3-unit rod can be slid along the 7.5-unit rod as the hinge angles vary, while a line can be drawn through the free end of the 7.5-unit rod parallel to the 3-unit rod. This line is side c of the major triangle.

    Reply
  3. ChrisC's avatar
    ChrisC Says:
    December 3, 2018 at 5:10 pm

    The c and the 3 are parallel, so 7.5/a=b/2=c/3. 2 equations and 3 unknowns so not suluble. However, if the angle at the top is a right angle (which it appears to be but is not so labelled), Pythagoras comes to our aid; a=root(5); b= 15/root(5); c=22.5/root(5)

    Reply
  4. John Peacock's avatar
    John Peacock Says:
    December 4, 2018 at 9:56 am

    I don’t think it can be taken as obvious that the angle at the top is exactly a right angle. But I do think it’s clear that the angle between b and c is to be taken as less than a right angle. In that case, the cosine rule allows 1 < a < sqrt(13), combined with b=15/a and c=3b/2.

    Reply
  5. Michel C.'s avatar
    Michel C. Says:
    December 4, 2018 at 6:58 pm

    We could print it and measure the angles experimentally… The angle at the top is clearly larger than 90 degrees. Ah, theorists…!

    Reply
  6. stallphill's avatar
    stallphill Says:
    December 4, 2018 at 8:38 pm

    It looks like you have to solve a cubic expression. Do it numerically.

    Reply
  7. Ilian Iliev's avatar
    Ilian Iliev Says:
    December 5, 2018 at 2:02 pm

    I quickly did it numerically and got a=2.02183658 b=7.41899723 and c= 11.12849584.

    Seems a tricky problem to give in school, though.

    Reply
  8. Ilian Iliev's avatar
    Ilian Iliev Says:
    December 5, 2018 at 2:22 pm

    If anyone cares, in Python the solution is:

    def f(x):
    return np.array([x[1]**2-x[3]**2-7.5**2+(x[2]-x[3])**2,2**2-x[4]**2-x[0]**2+(3.-x[4])**2,x[3]/x[4]-(x[2]-x[3])/(3.-x[4]),7.5/x[0]-x[1]/2.,x[3]/x[4]-x[1]/2.])

    guess=np.array([3.,3.,5.,1.,1.])

    roots_solve=scipy.optimize.fsolve(f,guess)

    print(roots_solve)

    [ 2.02183658 7.41899723 11.12849584 5.50995127 1.48536281]

    Reply

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