We are given a right triangle $\triangle ABC$ where $\angle BAC = 90^\circ$. Several points are defined as follows:

• $X$ is the foot of the altitude from $A$ to $BC$,
• $Y$ is the point where $\overline{AY}$ bisects $\angle BAC$,
• $Z$ is the point where $\overline{AZ}$ is the median from $A$ to the midpoint of $BC$.

Additionally, we are told that $\angle XAY = 22^\circ$, and we need to determine the measure of $\angle ZAC$.

Step 1: Relationship of $\angle BAC$

Since $\triangle ABC$ is a right triangle and $\angle BAC = 90^\circ$, we can analyze the triangle based on its properties. In any triangle, the median from the right angle vertex divides the hypotenuse into two equal segments. Moreover, the angle bisector and the altitude introduce further geometric relationships.

Step 2: Identifying Key Angles

The line segment $\overline{AY}$ is the bisector of $\angle BAC$. Since $\angle BAC = 90^\circ$, the bisector $\overline{AY}$ divides it into two equal angles: $\angle BAY = \angle CAY = \frac{90^\circ}{2} = 45^\circ.$

Step 3: Investigating $\angle XAY$

We are given that $\angle XAY = 22^\circ$. $X$ is the foot of the altitude, and since $\overline{AY}$ is the bisector of $\angle BAC$, this introduces additional relationships between the angles. Given that $\angle XAY = 22^\circ$, we can conclude that the angles near $A$ are constrained by these values.

Step 4: Median Line $\overline{AZ}$

Next, we analyze the median line $\overline{AZ}$. Since $Z$ is the midpoint of $BC$, the median divides the triangle into two smaller triangles with equal area. The important point here is that the median in a right triangle makes a specific angle with the other sides.

Step 5: Using Angle Relationships

The key fact we can use is that in a right triangle, the median from the right angle vertex makes a $45^\circ$ angle with the hypotenuse. Since $\overline{AZ}$ is the median, the angle $\angle ZAC$ is half of $\angle BAC$, which is $45^\circ$.

Thus, the measure of $\angle ZAC$ is $45^\circ$.

$\boxed{45^\circ}$

Would you like more details, or have any questions about this approach?

Here are five related questions you might explore:

1. How is the median of a right triangle calculated geometrically?
2. What are the properties of angle bisectors in triangles?
3. How does the altitude affect the properties of right triangles?
4. Can the sum of angles in a triangle be derived from the median and bisector properties?
5. How do angle bisectors relate to incenter and circumcenter properties?

Tip: Remember that the median to the hypotenuse in a right triangle is always half the length of the hypotenuse.