P, Q and R are three collinear points. P and Q are (3, 4) and (7, 7) respectively, and PR =10 units. Find the coordinates of R.

Answer

To find the coordinates of point R given that points P, Q, and R are collinear, we can follow these steps:

### Step 1: Identify the given points and information
- Points P and Q are given as:
- \( P(3, 4) \)
- \( Q(7, 7) \)
- The distance \( PR = 10 \) units.

### Step 2: Calculate the distance \( PQ \)
Using the distance formula:
\[ PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of P and Q:
\[ PQ = \sqrt{(7 - 3)^2 + (7 - 4)^2} \]
Calculating:
\[ PQ = \sqrt{(4)^2 + (3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \text{ units} \]

### Step 3: Determine the relationship between distances
Since \( PQ = 5 \) units and \( PR = 10 \) units, we can conclude that point Q is the midpoint of segment PR. This is because the distance from P to Q is half the distance from P to R.

### Step 4: Use the midpoint formula to find coordinates of R
Let the coordinates of R be \( R(x_3, y_3) \). The midpoint formula states:
\[ Q = \left( \frac{x_1 + x_3}{2}, \frac{y_1 + y_3}{2} \right) \]
Substituting the known values:
\[ (7, 7) = \left( \frac{3 + x_3}{2}, \frac{4 + y_3}{2} \right) \]

### Step 5: Set up equations for x and y coordinates
From the x-coordinates:
\[ 7 = \frac{3 + x_3}{2} \]
Multiplying both sides by 2:
\[ 14 = 3 + x_3 \]
Solving for \( x_3 \):
\[ x_3 = 14 - 3 = 11 \]

From the y-coordinates:
\[ 7 = \frac{4 + y_3}{2} \]
Multiplying both sides by 2:
\[ 14 = 4 + y_3 \]
Solving for \( y_3 \):
\[ y_3 = 14 - 4 = 10 \]

### Step 6: Write the coordinates of R
Thus, the coordinates of point R are:
\[ R(11, 10) \]

### Conclusion
Therefore, the coordinates of point R are \( (11, 10) \).

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