If segment BD has a measure of x which of the following expresses the

If segment BD has a measure of x which of the following expresses the perimeter of the triangle ABC in terms of x?


A. \(6x + 2x\sqrt{3}\)

B. \(6x + \frac{2x\sqrt{3}}{3}\)

C. \(4x + 4x\sqrt{3}\)

D. \(3x + x\sqrt{3}\)

E. \(2x + 2x\sqrt{3}\)

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Attachment:

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Solution:

• In triangle BCD,

o \(∠BDC = 90\)
o BCD is a 30: 60 : 90 triangle
o \(CD: BD: BC = 1: \sqrt{3}: 2\)

 \(\frac{BD}{BC} = \frac{\sqrt{3}}{2}\)
 \(\frac{x}{BC} = \frac{\sqrt{3}}{2}\)
 \(BC = \frac{2x }{ \sqrt{3}} = \frac{2*\sqrt{3}*x}{3}\)

• In triangle ABC,

o \(∠ABC = 90, ∠BCA = 60, ∠BAC = 30\)
o ABC is a 30: 60: 90 triangle
o \(BC: AB: AC = 1: \sqrt{3}: 2\)
o \(\frac{BC}{AB} = \frac{1}{\sqrt{3}}\)

 \(\frac{2*\sqrt{3}*x}{3*AB} = \frac{1}{(\sqrt{3})}\)
 \(AB = 2*x\)
o \(\frac{BC}{AC} = \frac{1}{ 2}\)

 \(\frac{2*\sqrt{3}*x}{3*AC} = \frac{1 }{ 2}\)
 \(AC = \frac{4*x*\sqrt{3}}{3}\)
• Perimeter of triangle = \(AB + BC + AC = 2*x + \frac{2*x*\sqrt{3}}{3} + \frac{4*\sqrt{3}*x}{3} = 2x + \frac{6*x*\sqrt{3}}{3} = 2x + {2*x*\sqrt{3}\)