In triangle BCD, with angles B = 59° and C = 31°, if BC = 19.2 and BD = 8.93, what is the measure of DC?

Steps to Solve

1. Identify the known values and angles We have the following information:

• Angle $B = 59^\circ$
• Angle $C = 31^\circ$
• Side $BC = 19.2$
• Side $BD = 8.93$

2. Calculate the missing angle To find angle $D$, we use the triangle angle sum property, which states that the sum of angles in a triangle is $180^\circ$.
   Thus,
   $$ D = 180^\circ - B - C $$
   Calculating this gives:
   $$ D = 180^\circ - 59^\circ - 31^\circ = 90^\circ $$

3. Apply the Law of Sines Now we can use the Law of Sines to find the length of side $DC$. The Law of Sines states:
   $$ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} $$
   For our triangle:

• $a = DC$
• $b = AC$ (which we can find using the Law of Sines)
• $c = BD = 8.93$

4. Set up the ratio using the Law of Sines Using the Law of Sines, we write:
   $$ \frac{DC}{\sin B} = \frac{BD}{\sin C} $$
   Substituting the known values:
   $$ \frac{DC}{\sin(59^\circ)} = \frac{8.93}{\sin(31^\circ)} $$

5. Solve for $DC$ Now we can isolate $DC$:
   $$ DC = \frac{8.93 \cdot \sin(59^\circ)}{\sin(31^\circ)} $$
   Using a calculator to find the sine values will provide the length of $DC$.