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Another Point In A Square (Posted on 2004-06-11) Difficulty: 3 of 5
There is a point M inside a square ABCD such that angle MAB is 60 and angle MCD is 15. Find angle MBC.

See The Solution Submitted by Brian Smith    
Rating: 3.2500 (4 votes)

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re: Solution (unproven) | Comment 3 of 16 |
(In reply to Solution (unproven) by Jer)

When all else fails, I prove analytically:

Orient the square with A=(0,0) B=(0,1) C=(1,1) D=(1,0)

The line through A has equation y=(sqrt(3)/3)x

The line through C has equation y=(2+sqrt(3))x - (1+sqrt(3))

Solving this system yields (after quite a bit of algebra)

x=sqrt(3)/2 and y=1/2

This implies MA = 1 = AB so triangle MAB is isosceles with vertex A = 60 degrees.   The two base angles are then also 60 degrees and since MBA = 60 and ABC = 90, MBC = 30.

-Jer 


  Posted by Jer on 2004-06-11 09:32:00
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