The problem states that circle has center $D$. The measure of angle $BAC$ is $38^\circ$. We need to find the measure of arc $BC$ and the measure of angle $BDC$.

1. Problem Description

The problem states that circle has center $D$. The measure of angle $BAC$ is $38^\circ$. We need to find the measure of arc $BC$ and the measure of angle $BDC$.

2. Solution Steps

(a) We are given that $m\angle BAC = 38^\circ$. Angle $BAC$ is an inscribed angle intercepting arc $BC$. The measure of an inscribed angle is half the measure of its intercepted arc.

Therefore, $m\angle BAC = \frac{1}{2} m\stackrel{\frown}{BC}$.

Multiplying both sides by 2, we have $m\stackrel{\frown}{BC} = 2 \cdot m\angle BAC$.

Substituting $m\angle BAC = 38^\circ$, we get $m\stackrel{\frown}{BC} = 2 \cdot 38^\circ = 76^\circ$.

(b) We are given that $m\angle BAC = 38^\circ$. Since $D$ is the center of the circle, $\angle BDC$ is a central angle intercepting arc $BC$. The measure of a central angle is equal to the measure of its intercepted arc.

Therefore, $m\angle BDC = m\stackrel{\frown}{BC}$.

From part (a), we found that $m\stackrel{\frown}{BC} = 76^\circ$.

Thus, $m\angle BDC = 76^\circ$.

3. Final Answer

(a) $m\stackrel{\frown}{BC} = 76^\circ$

(b) $m\angle BDC = 76^\circ$