The problem asks to find the measure of arc $DC$ given that the measure of inscribed angle $\angle BDC = 38^\circ$ and the central angle $\angle BOC = 70^\circ$. The dot represents the center of the circle.

1. Problem Description

The problem asks to find the measure of arc $DC$ given that the measure of inscribed angle $\angle BDC = 38^\circ$ and the central angle $\angle BOC = 70^\circ$. The dot represents the center of the circle.

2. Solution Steps

First, we relate the measure of an inscribed angle to the measure of the intercepted arc. The inscribed angle theorem states that the measure of an inscribed angle is half the measure of its intercepted arc. In this case, the inscribed angle $\angle BDC$ intercepts arc $BC$. Therefore,

$m\angle BDC = \frac{1}{2} m\text{arc } BC$

$38^\circ = \frac{1}{2} m\text{arc } BC$

$m\text{arc } BC = 2 \times 38^\circ = 76^\circ$

Next, we know that the measure of a central angle is equal to the measure of its intercepted arc. So, $m\angle BOC = m\text{arc } BC$.

However, we are given that $m\angle BOC = 70^\circ$, so $m\text{arc } BC = 70^\circ$. This value contradicts the previous value, which we calculated to be $76^\circ$. There seems to be an error, as the inscribed angle $\angle BDC$ and the central angle $\angle BOC$ intercept the same arc, and they should be $38^\circ$ and $76^\circ$, respectively. However, they are given as $38^\circ$ and $70^\circ$. We proceed with the given value, $70^\circ$.

We need to find the measure of arc $DC$. The measure of the whole circle is $360^\circ$. Thus, $m\text{arc } DB + m\text{arc } BC + m\text{arc } CD = 360^\circ$. We are also given that $\angle BDC = 38^\circ$. Since the inscribed angle is $38^\circ$, the arc $BC$ is $76^\circ$. The arc $CD$ can be found from the central angle $\angle COD$, which is twice the inscribed angle $\angle CBD$. However, the inscribed angle $\angle CBD$ is unknown.

Since the measure of arc $BC$ is $70^\circ$, the central angle subtended by the arc BC is $\angle BOC = 70^\circ$.

Since the total angle is $360^\circ$, we can say that the arc $BDC$ can be calculated by $m\text{arc } BDC = 360^\circ - m\text{arc } CB$.

Also, the measure of the inscribed angle $\angle BDC = 38^\circ$ intercepts the arc $BC$, so $m\text{arc } BC = 2 \times 38 = 76^\circ$. We also know that $m\text{arc } BC = 70^\circ$, but we proceed assuming the $m\text{arc } BC = 70^\circ$.

Then we can find the $m\text{arc } BDC = 360^\circ - 70^\circ = 290^\circ$.

We need to find $m\text{arc } DC = m\text{arc } BDC - m\text{arc } BD$.

Since we don't have enough information to find $m\text{arc } BD$, we will look at another approach.

Let $x = m\text{arc } DC$. The measure of $\angle BDC = 38^\circ = \frac{1}{2} \times m\text{arc } BC$, and $m\text{arc } BC = 76^\circ$. Also, $m\text{arc } BC = 70^\circ$.

If the central angle is $70^\circ$, and $m \angle BDC = 38^\circ$. The measure of the whole circle is $360^\circ$.

The arc $BC = 70^\circ$. Then the arc $BD + arc DC = 360 - 70 = 290$. Let the arc $DC = x$.

$m\angle BDC = 38^\circ$, the measure of the arc $BC = 2\times 38 = 76^\circ$. We have two different measurements, but assuming $m\text{arc } BC = 70^\circ$

Central angle $70^\circ$

$m\text{arc } DC$. Let it be $x$.

Inscribed angle $m\angle BDC = 38^\circ$.

Central $\angle BOC = 70^\circ = m\text{arc } BC$.

Let us assume that inscribed $\angle DBC = y$ then the arc it subtends is $DC = 2y = x$.

$\angle BDC = 38^\circ$.

Now the problem is under determined.

3. Final Answer

I am unable to determine the measure of the arc $DC$ with the information given.