Given is a Right triangle ΔACB with angle ∠ACB = 90 degrees.
Given that CD⊥AB, it means we have two Right triangles ΔCDA and ΔCDB.
Given angle ∠ACD = 30 degrees, it means ∠A = 60° and ∠B = 30°
Given side AC = 6 centimeters.
It says to find BD = ?
In Right triangle ΔCDA; CD is adjacent, AC is hypotenuse, ∠ACD = 30°
[tex] Cos(\angle ACD)=\frac{adjacent}{hypotenuse} \\\\Cos(30^o)=\frac{CD}{AC} \\\\\frac{\sqrt{3}}{2} =\frac{CD}{6} \\\\CD = 3\sqrt{3} \;cm [/tex]
In Right triangle ΔCDB; CD is opposite, BD is adjacent, ∠B = 30°
[tex] Tan(\angle B) = \frac{opposite}{adjacent} \\\\Tan(30^o) = \frac{CD}{BD} \\\\\frac{1}{\sqrt{3}} = \frac{3\sqrt{3}}{BD} \\\\Cross \;\;multiplying \\\\BD = \sqrt{3} * 3\sqrt{3} \\\\BD = 9 \;cm [/tex]
Hence, final answer is BD = 9 centimeters.
