P(-3, 7) and Q(1, 9) are two points. Find the point R on PQ such that `PR:QR = 1:1`.

Answer

To find the point R on the line segment PQ such that the ratio PR:QR = 1:1, we can use the section formula. Here are the steps to solve the problem:

### Step 1: Identify the coordinates of points P and Q
The coordinates of point P are given as P(-3, 7) and the coordinates of point Q are Q(1, 9).

### Step 2: Understand the ratio
Since PR:QR = 1:1, it means that point R divides the line segment PQ into two equal parts. Therefore, R is the midpoint of PQ.

### Step 3: Use the midpoint formula
The midpoint (R) of a line segment connecting two points (x1, y1) and (x2, y2) is given by the formula:
\[ R\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]

### Step 4: Substitute the coordinates of P and Q into the midpoint formula
Here, we have:
- \(x_1 = -3\), \(y_1 = 7\) (coordinates of P)
- \(x_2 = 1\), \(y_2 = 9\) (coordinates of Q)

Now substituting these values into the midpoint formula:
\[ R\left(\frac{-3 + 1}{2}, \frac{7 + 9}{2}\right) \]

### Step 5: Calculate the x-coordinate of R
\[ R_x = \frac{-3 + 1}{2} = \frac{-2}{2} = -1 \]

### Step 6: Calculate the y-coordinate of R
\[ R_y = \frac{7 + 9}{2} = \frac{16}{2} = 8 \]

### Step 7: Write the coordinates of point R
Thus, the coordinates of point R are:
\[ R(-1, 8) \]

### Final Answer:
The point R on the line segment PQ such that PR:QR = 1:1 is R(-1, 8).

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