As shown in the diagram $\angle BAC=90^\circ$. Let $X$ be the foot of the altitude from $A$ to $\overline {BC}$, $\overline{AY}$ be the bisector of $\angle BAC$, and $\overline{AZ}$ be the median of $\triangle ABC$. If $\angle XAY = 22^\circ$, then what is the measure of $\angle ZAC$ in degrees?

Given the problem, we know the following information:

- $\angle BAC = 90^\circ$.
- $X$ is the foot of the altitude from $A$ to $\overline{BC}$.
- $\overline{AY}$ is the bisector of $\angle BAC$.
- $\overline{AZ}$ is the median of $\triangle ABC$.
- $\angle XAY = 22^\circ$.

We are required to find the measure of $\angle ZAC$.

### Step 1: Understanding the Problem
The triangle $ABC$ is a right triangle with $\angle BAC = 90^\circ$. $X$ is the foot of the altitude from $A$ to the hypotenuse $BC$, so $\overline{AX}$ is perpendicular to $\overline{BC}$. The line $\overline{AY}$ bisects $\angle BAC$, so it divides the $90^\circ$ angle into two equal angles, each of $45^\circ$.

### Step 2: Calculation of $\angle YAX$
Since $\angle YAX$ is one part of $\angle BAC$, and the angle bisector divides $\angle BAC = 90^\circ$ into two equal parts:

\[
\angle BAY = 45^\circ
\]

Given that $\angle XAY = 22^\circ$, we can calculate $\angle YAX$ as:

\[
\angle YAX = \angle BAY - \angle XAY = 45^\circ - 22^\circ = 23^\circ
\]

### Step 3: Analysis of Median and Related Angles
$AZ$ is the median of the triangle, so $Z$ is the midpoint of $BC$. In a right triangle, the median to the hypotenuse is equal to half the hypotenuse and forms an angle of $45^\circ$ with the legs of the triangle. However, since $AZ$ is not directly along the leg but along the hypotenuse, $\angle ZAC$ is not directly a simple angle but can be derived using trigonometric properties and the given angles.

### Step 4: Determine $\angle ZAC$
In triangle $ABC$, the total angle at $A$ must sum up considering the median and altitude as:

\[
\angle ZAC = \angle YAX = 23^\circ
\]

This is based on the fact that the median, in this case, bisects the remaining angle.

### Conclusion
Thus, the measure of $\angle ZAC$ is $ \boxed{23^\circ} $.

---

Would you like further details on this solution or have any questions? Here are some related questions:

1. How does the altitude affect other angles in a right triangle?
2. What is the significance of the angle bisector in this problem?
3. How do you calculate the angles in a triangle with given side lengths?
4. What are the properties of the median in a right triangle?
5. How does the hypotenuse's median relate to the triangle's angles?
6. What role does the perpendicular foot from a vertex play in angle calculation?
7. How do you solve similar problems where only partial angle measures are given?
8. What trigonometric functions can simplify the angle calculations in right triangles?

**Tip:** When dealing with right triangles, always consider the special properties of medians, altitudes, and bisectors, as they often provide shortcuts to finding unknown angles or sides.

Explore how to solve a triangle angle problem involving altitude, angle bisector, and median. Given that angle BAC is 90 degrees, find the measure of angle ZAC when angle XAY is 22 degrees. Includes step-by-step explanation and solution approach.

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