Let BD=x.
Right triangle ABD
BD=ABtanα,AD=cosαABGiven AB=2,BC=4.
Substitute
x=2tanα,AD=cosα2 Since ED∥AB, then right triangles ABC and ABD are similar.
ABED=BCCD=>ED=BCAB⋅CD
CD=BC−BD=4−2tanα Substitute
ED=42(4−2tanα)=2−tanα Then
L=AD+ED=cosα2+2−tanα,0°<α<90° Find the first derivative with respect to α
Lα′=−cos2α2(−sinα)−cos2α1
=cos2α2sinα−1 Find the critical number(s)
Lα′=0=>cos2α2sinα−1=0
sinα=21,0°<α<90°The critical number is 30°.
If 0°<α<30°,Lα′<0,L decreases.
If 30°<α<90°,Lα′>0,L increases.
The function L has a local minimum at α=30°.
Since the function L has the only extremum for 0°<α<90°, then the function L has the absolute minimum for 0°<α<90° at α=30°.
∠BAD=α=30°.
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