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Distance to Diagonal 2 (Posted on 2013-06-06) Difficulty: 2 of 5

Let ABCD be a parallelogram with ∠BAD = 45° and ∠ABD = 30°.

What is the distance from B to diagonal AC in terms of just |AB|?

What is the distance from B to diagonal AC in terms of just |AD|?

See The Solution Submitted by Bractals    
Rating: 4.0000 (1 votes)

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Solution Solution Comment 1 of 1
It feels as if there should be a simple geometry proof here.
I can’t find one - so here’s some trig:

Denote the lengths of AB, AD by x and y, and the angle CAD by Z.
Let p be the length of the perpendicular from B to AC.

Using the Sine Rule in triangles ABC and ABD:

y/x = sin(45o – Z)/sin Z  = sin 30o/sin 105o

Solving for Z:   sin Z = sin(45o – Z) sin(45o + 60o)/sin 30o

            2 sin Z = (cos Z – sin Z)(1 + sqrt3)

            tan Z = (1 + sqrt3) / (sqrt3 + 3)  = 1 / sqrt3

            Z  = 30o

So                    p = y sin 30o  =  |AD|/2

and                   p = x sin 15o = |AB|sqrt2(sqrt3 - 1)/4 



  Posted by Harry on 2013-06-14 16:34:52
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