Triangle Segment Lengths Using Angle Bisector Theorem

In triangle ABC, angle A is 30 degrees, and angle B is 60 degrees. The length of side AC is 10 units. Let D be the point on side BC such that AD bisects angle A. Determine the lengths of segments BD and DC.

Use the Angle Bisector Theorem, which states that if a point D is on side BC of triangle ABC, then the ratio of the lengths of the segments BD and DC is equal to the ratio of the lengths of the sides opposite those segments. Mathematically, this can be expressed as:

$\f\frac{BD}{DC} = \f\frac{AB}{AC}$

To solve, first, calculate the length of side AB using the law of sines:

$\f\frac{AB}{\sin C} = \f\frac{AC}{\sin B}$

Find angle C by subtracting angles A and B from 180 degrees. Finally, use the values for segments BD and DC in terms of x, where $x = BD$ and proceed to calculate.