Solutions – sum of the angles
December 27, 2007 am31 10:20 am
(Way cool that this image is fine for illustrating my problem, but is really from a fairly advanced seeming math blog (on game theory?), on a topic that I do not understand: Bosker Blog.
Consider rectangle ADEH with side ABCD and side EFGH such that AB = BC = CD = DE = EF = FG = GH = HA.
Find the sum of ∠FAD, ∠GAD, and ∠HAD ∠EAD, ∠FAD, and ∠GAD.
Answers in the comments section, above.
Questions? Over here.
9 Comments
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JD2718 
∠GAD is clearly pi/4. Some simple trig shows that ∠FAD = arctan(1/2) and ∠EAD = arctan(1/3), so we want to find arctan(1/2) + arctan(1/3). Using the sum formula for tangent:
tan(arctan(1/2) + arctan(1/3)) = (1/2 + 1/3)(1 – (1/2)(1/3)) = (5/6)/(5/6) = 1.
So ∠EAD + ∠FAD = arctan(1) = pi/4, and the sum of all three is pi/2. Astounding!
I’m still not satisfied with this solution, though, since it seems like there ought to be a nice *geometric* explanation of why these three angles add up to a right angle, and I’m not seeing it…
A solution without trig would be nice, especially since the answer looks so simple….
If you accept Pythagoras as a “geometric explanation” then it’s easy. Label the three angles, largest to smallest, as “a,” “b,” and “c.”
As noted, a is 45deg.
b is the smallest angle in a 1 – 2 – rt(5) triangle.
c is the smallest angle in a 1 – 3 – rt(10) triangle.
Construct a triangle D(-2,1) E(1,2) F(0,0) and see that D = b+c and that it appears to be a right triangle with right angle F.
Use slopes to show that the legs DF and EF are perpendicular, i.e., m1=-1/m2
Distance formula gives you two equal legs.
An isosceles right triangle has angles of 45-45-90
and each base angle is b + c = 45
so a+b+c = 90
Nice diagram. You’ve used up both of my non-trig solutions!
the easiest way I see is to draw the lattice points (0,0), (4,2), and (3,-1). if you call angles EAD, FAD, and GAD X, Y, and Z respectively, you see that you get an isoceles triangle where one of the angles is X+Y and the other angle is 90 – X – Y, so X+Y = 45, and X+Y+Z = 90.
it also looks like I am 3 weeks late for the carnival. =)