Question

In ΔАВС, m∠ACB = 90°, CD ⊥ AB m∠ACD = 30°, and AC = 6 cm. Find BD

Answer 1

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°

$Cos(\angle ACD)=\frac{adjacent}{hypotenuse} \\\\Cos(30^o)=\frac{CD}{AC} \\\\\frac{\sqrt{3}}{2} =\frac{CD}{6} \\\\CD = 3\sqrt{3} \;cm$

In Right triangle ΔCDB; CD is opposite, BD is adjacent, ∠B = 30°

$Tan(\angle B) = \frac{opposite}{adjacent} \\\\Tan(30^o) = \frac{CD}{BD} \\\\\frac{1}{\sqrt{3}} = \frac{3\sqrt{3}}{BD} \\\\Cross \;\;multiplying \\\\BD = \sqrt{3} * 3\sqrt{3} \\\\BD = 9 \;cm$

Hence, final answer is BD = 9 centimeters.

Answer 2

It is 9 because you have a 30,60,90 triangle, and then you can use pythagorean theorem.