Solve for R in terms of A, P, and x:

Given:

A = H + y

P = W + x

H = C + z

R = A - C

R = (y^2 + x^2) / 2y

P^2 + C^2 = R^2 + H^2 + W^2

R^2 = z^2 + x^2

Tan(theta) = x / z = H / W

x / z = -H / W

To solve for \( R \) in terms of \( A \), \( P \), and \( x \) using the given equations, let's proceed step by step.

### Step 1: List the given equations
1. \( A = H + y \)
2. \( P = W + x \)
3. \( H = C + z \)
4. \( R = A - C \)
5. \( R = \frac{y^2 + x^2}{2y} \)
6. \( P^2 + C^2 = R^2 + H^2 + W^2 \)
7. \( R^2 = z^2 + x^2 \)
8. \( \tan(\theta) = \frac{x}{z} = \frac{H}{W} \)
9. \( \frac{x}{z} = -\frac{H}{W} \)

### Step 2: Simplify and analyze

**From equation (4):**  
\[ R = A - C \]
Substitute \( A = H + y \) and \( H = C + z \) from equations (1) and (3):
\[ R = (C + z + y) - C = z + y \]

So, 
\[ R = z + y \tag{10} \]

**From equation (7):**  
\[ R^2 = z^2 + x^2 \]

Substitute \( R = z + y \) from equation (10) into this equation:
\[ (z + y)^2 = z^2 + x^2 \]
Expanding:
\[ z^2 + 2zy + y^2 = z^2 + x^2 \]
\[ 2zy + y^2 = x^2 \]

This equation can be factored:
\[ y(2z + y) = x^2 \]
\[ 2z + y = \frac{x^2}{y} \tag{11} \]

**From equation (5):**  
\[ R = \frac{y^2 + x^2}{2y} \]

Substitute equation (10) into this:
\[ z + y = \frac{y^2 + x^2}{2y} \]

Multiply both sides by \( 2y \):
\[ 2yz + 2y^2 = y^2 + x^2 \]
\[ 2yz + y^2 = x^2 \]

Notice this equation matches equation (11), confirming that:
\[ z + y = \frac{y^2 + x^2}{2y} \]

### Step 3: Relate \( R \) to \( P \) and \( x \)

Since \( R = z + y \) and using \( z = \frac{x^2 - y^2}{2y} \) (derived from the above):
\[ R = \frac{x^2}{2y} + y \]

**From equation (2):**  
\[ P = W + x \]
\[ W = P - x \tag{12} \]

**From equation (8):**
\[ \frac{x}{z} = \frac{H}{W} \]
Since \( z + y = R \):
\[ \frac{x}{R - y} = \frac{H}{W} \]
\[ W = \frac{H(R - y)}{x} \tag{13} \]

Using \( W = P - x \) in equation (13):
\[ P - x = \frac{H(R - y)}{x} \]

Finally, solving for \( R \):
\[ R = \frac{xP + yW}{x} \]

Since \( W = P - x \), substitute back:
\[ R = \frac{xP + y(P - x)}{x} \]

Now, let's simplify:
\[ R = P + \frac{y(P - x)}{x} \]

### Final expression for \( R \):

\[ R = P \left( 1 + \frac{y}{x} \right) - y \]

### Conclusion:
\[ R = \frac{P^2 + x^2 - C^2}{2P} \]

This is an expression of \( R \) in terms of \( P \), \( x \), and \( y \).

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Would you like further details on any specific part of the solution? Here are some related questions:

1. How would changing the value of \( y \) affect the value of \( R \)?
2. Can \( R \) be simplified further if \( y = 0 \)?
3. How does the equation change if \( C = 0 \)?
4. What are the implications of the negative tangent condition \( \frac{x}{z} = -\frac{H}{W} \)?
5. Can \( P \) be solved for in terms of \( R \), \( x \), and \( C \)?

**Tip:** When solving for a variable, substitute all given relationships step by step to eliminate others.

Explore a detailed mathematical solution for finding R in terms of A, P, and x. This problem involves algebraic equations and trigonometric identities, covering relationships between variables and step-by-step derivations. Suitable for advanced high school students.

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