P, Q, R are three collinear points. The coordinates of P and R are (3, 4) and (11, 10) respectively, and PQ is equal to 2.5 units. Coordinates of Q are

Options

A. (5, 11/2)
B. (11, 5/2)
C. (5, -11/2)
D. (-5, 11/2)

Answer

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

### Step 1: Identify the coordinates of points P and R.
- The coordinates of point P are (3, 4).
- The coordinates of point R are (11, 10).

### Step 2: Calculate the distance PR.
Using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of P and R:
\[ d = \sqrt{(11 - 3)^2 + (10 - 4)^2} \]
Calculating:
\[ d = \sqrt{(8)^2 + (6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \]
So, the distance PR is 10 units.

### Step 3: Find the midpoint S of segment PR.
The midpoint formula is:
\[ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Substituting the coordinates of P and R:
\[ S = \left( \frac{3 + 11}{2}, \frac{4 + 10}{2} \right) = \left( \frac{14}{2}, \frac{14}{2} \right) = (7, 7) \]

### Step 4: Calculate the distance PS.
Using the distance formula again:
\[ PS = \sqrt{(7 - 3)^2 + (7 - 4)^2} \]
Calculating:
\[ PS = \sqrt{(4)^2 + (3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

### Step 5: Determine the distance PQ.
Given that PQ is equal to 2.5 units, we can find the position of Q relative to P and S.

### Step 6: Find the coordinates of Q.
Since Q lies on the line segment PS, we can use the section formula. The distance from P to S is 5 units, and the distance from P to Q is 2.5 units. This means Q divides PS in the ratio 1:1 (because 2.5 is half of 5).

Using the section formula:
\[ Q = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]
where \( m = 1 \), \( n = 1 \), \( (x_1, y_1) = (3, 4) \), and \( (x_2, y_2) = (7, 7) \):
\[ Q = \left( \frac{1 \cdot 7 + 1 \cdot 3}{1 + 1}, \frac{1 \cdot 7 + 1 \cdot 4}{1 + 1} \right) = \left( \frac{7 + 3}{2}, \frac{7 + 4}{2} \right) = \left( \frac{10}{2}, \frac{11}{2} \right) = (5, 5.5) \]

### Conclusion:
The coordinates of point Q are (5, 5.5).

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