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.
+9466 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 = \(2\sqrt3-2\)
