The coordinates of P and Q are (- 3, 4) and (2, 1), respectively. If PQ is extended to R such that PR = 2QR, then what are the coordinates of R ?

Options

A. (3, 7)
B. (2, 4)
C. `(-(1)/(2),(5)/(2))`
D. (7,-2)

Answer

To find the coordinates of point R, we will follow these steps:

### Step 1: Understand the relationship between PR and QR
We are given that PR = 2QR. This means that the segment PR is twice as long as the segment QR. If we let QR be represented as 'd', then PR would be '2d'. Therefore, the entire length from P to R can be expressed as:
\[ PR = PQ + QR = PQ + d = 2d \]
This implies that:
\[ PQ = d \]
Thus, point Q divides the segment PR into two equal parts.

### Step 2: Calculate the coordinates of Q
The coordinates of points P and Q are given as:
- P = (-3, 4)
- Q = (2, 1)

### Step 3: Use the midpoint formula
Since Q divides PR into two equal halves, we can use the midpoint formula. The midpoint M of a segment with endpoints (x1, y1) and (x2, y2) is given by:
\[ M = \left( \frac{x1 + x2}{2}, \frac{y1 + y2}{2} \right) \]
Here, we can set Q as the midpoint of P and R.

Let the coordinates of R be (x, y). Then, we can set up the equations based on the midpoint formula:
\[ Q = \left( \frac{-3 + x}{2}, \frac{4 + y}{2} \right) \]

### Step 4: Set up equations from the midpoint
From the coordinates of Q (2, 1), we can set up the following equations:
1. For the x-coordinates:
\[ 2 = \frac{-3 + x}{2} \]
2. For the y-coordinates:
\[ 1 = \frac{4 + y}{2} \]

### Step 5: Solve for x
From the first equation:
\[ 2 = \frac{-3 + x}{2} \]
Multiplying both sides by 2:
\[ 4 = -3 + x \]
Adding 3 to both sides:
\[ x = 7 \]

### Step 6: Solve for y
From the second equation:
\[ 1 = \frac{4 + y}{2} \]
Multiplying both sides by 2:
\[ 2 = 4 + y \]
Subtracting 4 from both sides:
\[ y = -2 \]

### Step 7: Write the coordinates of R
Thus, the coordinates of point R are:
\[ R = (7, -2) \]

### Final Answer
The coordinates of R are (7, -2).