The middle point of the segment of the straight line joining the points (p,q) and (q,-p) is (r/2 , s/2) . What is the length of the segment ?

Options

A. `[(s^(2) + r + 2)^(1//2) ]//2`
B. `[(s^(2) + r^(2))^(1//2) ]//4`
C. `(s^(2) + r^(2))^(1//2)`
D. `s + r`

Answer

To find the length of the segment joining the points (p, q) and (q, -p), we can follow these steps:

### Step 1: Identify the Points
The two points given are:
- Point A: (p, q)
- Point B: (q, -p)

### Step 2: Use the Midpoint Formula
The formula for the midpoint M of a line segment joining two points (x1, y1) and (x2, y2) is:
\[ M = \left( \frac{x1 + x2}{2}, \frac{y1 + y2}{2} \right) \]
For our points:
- \(x1 = p\), \(y1 = q\)
- \(x2 = q\), \(y2 = -p\)

Calculating the midpoint:
\[ M = \left( \frac{p + q}{2}, \frac{q - p}{2} \right) \]

### Step 3: Set Midpoint Equal to Given Values
According to the problem, the midpoint is also given as \((\frac{r}{2}, \frac{s}{2})\). Therefore, we can set up the following equations:
1. \(\frac{p + q}{2} = \frac{r}{2}\)
2. \(\frac{q - p}{2} = \frac{s}{2}\)

### Step 4: Solve for p and q
From the first equation:
\[ p + q = r \quad \text{(1)} \]
From the second equation:
\[ q - p = s \quad \text{(2)} \]

### Step 5: Solve the System of Equations
We can solve equations (1) and (2) simultaneously. Adding equations (1) and (2):
\[ (p + q) + (q - p) = r + s \]
This simplifies to:
\[ 2q = r + s \implies q = \frac{r + s}{2} \quad \text{(3)} \]

Now substituting equation (3) back into equation (1):
\[ p + \frac{r + s}{2} = r \]
This simplifies to:
\[ p = r - \frac{r + s}{2} = \frac{2r - r - s}{2} = \frac{r - s}{2} \quad \text{(4)} \]

### Step 6: Calculate the Length of the Segment
The length of the segment between points A and B can be calculated using the distance formula:
\[ \text{Length} = \sqrt{(x2 - x1)^2 + (y2 - y1)^2} \]
Substituting the coordinates of points A and B:
\[ \text{Length} = \sqrt{(q - p)^2 + (-p - q)^2} \]

### Step 7: Substitute p and q
Using equations (3) and (4):
- Substitute \(p = \frac{r - s}{2}\) and \(q = \frac{r + s}{2}\):
\[ \text{Length} = \sqrt{\left(\frac{r + s}{2} - \frac{r - s}{2}\right)^2 + \left(-\frac{r - s}{2} - \frac{r + s}{2}\right)^2} \]
Calculating the first term:
\[ \frac{r + s - (r - s)}{2} = \frac{2s}{2} = s \]
Calculating the second term:
\[ -\frac{(r - s) + (r + s)}{2} = -\frac{2r}{2} = -r \]
Thus, the length becomes:
\[ \text{Length} = \sqrt{s^2 + r^2} \]

### Final Answer
The length of the segment is:
\[ \text{Length} = \sqrt{r^2 + s^2} \]