There is a point P in a rectangle ABCD, such that PA = 4, PD = 5, PB = 8, find PC.

Answer

To solve the problem of finding the length \( PC \) in rectangle \( ABCD \) with point \( P \) inside it, we can use the following steps:

### Step-by-Step Solution:

1. **Identify the Given Values**:
- \( PA = 4 \)
- \( PD = 5 \)
- \( PB = 8 \)
- We need to find \( PC \).

2. **Use the Formula**:
In a rectangle, the relationship between the distances from a point inside the rectangle to the vertices is given by the formula:
\[ PA^2 + PC^2 = PD^2 + PB^2 \]

3. **Substitute the Known Values**:
Substitute the known values into the formula:
\[ 4^2 + PC^2 = 5^2 + 8^2 \]

4. **Calculate the Squares**:
- Calculate \( 4^2 \):
\[ 4^2 = 16 \]
- Calculate \( 5^2 \):
\[ 5^2 = 25 \]
- Calculate \( 8^2 \):
\[ 8^2 = 64 \]

5. **Set Up the Equation**:
Substitute the calculated squares back into the equation:
\[ 16 + PC^2 = 25 + 64 \]

6. **Simplify the Right Side**:
Combine the numbers on the right side:
\[ 25 + 64 = 89 \]
So, the equation becomes:
\[ 16 + PC^2 = 89 \]

7. **Isolate \( PC^2 \)**:
Subtract 16 from both sides to isolate \( PC^2 \):
\[ PC^2 = 89 - 16 \]
\[ PC^2 = 73 \]

8. **Take the Square Root**:
To find \( PC \), take the square root of both sides:
\[ PC = \sqrt{73} \]

### Final Answer:
Thus, the length \( PC \) is \( \sqrt{73} \) cm.

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