In the figure, $$ △ABC $$ is a right triangle, where the internal angle $$ \angle C = 90\degree $$ , $$ \angle A=30° $$ , the segment $$ AD $$ bisects $$ ∠A $$ , which one is the value of the expression $$ \dfrac{AB}{CD}-\dfrac{AC}{CD} $$

Question

In the figure, $$ △ABC $$ is a right triangle, where the internal angle  $$ \angle C = 90\degree $$ , $$ \angle A=30° $$ ,   the segment $$ AD $$ bisects  $$ ∠A $$ , which one is  the value of the expression $$ \dfrac{AB}{CD}-\dfrac{AC}{CD} $$

Uniteasy Admin · Accepted Answer

Construct a perpendicular line from point D to AB and intersects AB at point D
 $$ DE\perp AB $$ .

$$ \dfrac{AB}{CD}-\dfrac{AC}{CD} $$
$$ =\dfrac{AB-AC}{CD}  $$
$$ =\dfrac{BE}{CD}  $$
$$ =\dfrac{BE}{DE}  $$

$$ \because  ∠DBE = 60\degree  $$

$$DB = 2BE$$
$$ DB^2=BE^2+DE^2 $$

$$ 4BE^2 = BE^2+DE^2 $$
$$ 3BE^2 = DE^2 $$

$$ 	herefore   \dfrac{BE}{DE} = \dfrac{\sqrt{3}  }{3} $$

Multiple Choice Question (MCQ)

Answer