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In the diagram, square ABCD has sides of length 4, and triangle ABE is equilateral. Line segments BC and AC intersect at P. Point Q is on BC so that PQ is perpendicular to BC and PQ=x.

Find the value of x in simplest radical form.

 Jun 29, 2019
 #1
avatar+8406 
+3

Notice that     m∠PBQ  =  90° - 60°  =  30°     so   △PBQ  is a  30-60-90 triangle. So

 

BQ  =  x√3

 

And notice     m∠PCQ  =  45°     so   △PCQ is a  45-45-90  triangle. So

 

QC  =  x

 

And

 

BQ + QC  =  4

                                  Substitute   x√3   in for  BQ   and   x   in for  QC

x√3 + x  =  4

                                  Factor  x  out of the terms on the left side.

x(√3 + 1)  =  4

                                  Divide both sides of the equation by   (√3 + 1)

x  =  \(\frac{4}{\sqrt3+1}\)

                                  Multiply the numerator and denominator by  √3 - 1  and simplify

x  =  \(\frac{4}{\sqrt3+1}\cdot\frac{\sqrt3-1}{\sqrt3-1}\)

 

x  =  \(\frac{4\sqrt3-4}{3-1}\)

 

x  =  \(\frac{4\sqrt3-4}{2}\)

 

x  =  \(2\sqrt3-2\)

.
 Jun 29, 2019

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