Parts of a Triangle: Using the theorem: The median of a triangle divides it into two triangles of the same area

To solve this problem, we'll employ the theorem which states that the median of a triangle divides it into two triangles of equal area. Given that $AD$ is a median of $\triangle ABC$, it divides $\triangle ABC$ into $\triangle ADB$ and $\triangle ADC$.

Step 1: Recognize the properties of a median. The median $AD$ implies:
- Area of $\triangle ADB =$ Area of $\triangle ADC$.
- Given Area of $\triangle ADB = 15$, hence Area of $\triangle ADC = 15$.

Step 2: Compute the total area of $\triangle ABC$:
- Total Area of $\triangle ABC =$ Area of $\triangle ADB +$ Area of $\triangle ADC = 15 + 15 = 30.$

Thus, the area of triangle $\triangle ABC$ is $\boxed{30}$.

To solve this problem, we'll use the property that a median of a triangle divides it into two triangles of equal area. Given the area of $\triangle ABC$ is 32:

• Since $AD$ is the median, it divides $\triangle ABC$ into two triangles $\triangle ABD$ and $\triangle ADC$ of equal area.
• Thus, the area of $\triangle ADC = \frac{1}{2} \times \text{Area of } \triangle ABC = \frac{1}{2} \times 32 = 16.$

Therefore, the area of triangle $\triangle ADC$ is $16$.