`P` and `Q` are points on the line joining `A(-2,5)` and `B(3,1)` such that `A P=P Q=Q B` . Then, the distance of the midpoint of `P Q` from the origin is 3 (b) `(sqrt(37))/2` (b) 4 (d) 3.5

Options

A. 3
B. `sqrt(37//2)`
C. 4
D. `3.5`

Explanation

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Identify the coordinates of points A and B
We have the points:
- \( A(-2, 5) \)
- \( B(3, 1) \)

### Step 2: Determine the midpoint of line segment AB
The midpoint \( M \) of a line segment connecting two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Substituting the coordinates of points A and B:
\[ M = \left( \frac{-2 + 3}{2}, \frac{5 + 1}{2} \right) = \left( \frac{1}{2}, 3 \right) \]

### Step 3: Understand the relationship between points P, Q, and A, B
Since \( AP = PQ = QB \), points P and Q must be positioned such that the segments AP, PQ, and QB are equal. This implies that the midpoint of segment PQ is the same as the midpoint of segment AB. Therefore, the midpoint of PQ is also \( M = \left( \frac{1}{2}, 3 \right) \).

### Step 4: Calculate the distance from the midpoint of PQ to the origin
To find the distance \( d \) from the midpoint \( M \left( \frac{1}{2}, 3 \right) \) to the origin \( O(0, 0) \), we use the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates:
\[ d = \sqrt{\left( \frac{1}{2} - 0 \right)^2 + (3 - 0)^2} = \sqrt{\left( \frac{1}{2} \right)^2 + 3^2} \]
Calculating further:
\[ d = \sqrt{\frac{1}{4} + 9} = \sqrt{\frac{1}{4} + \frac{36}{4}} = \sqrt{\frac{37}{4}} = \frac{\sqrt{37}}{2} \]

### Step 5: Conclusion
The distance of the midpoint of PQ from the origin is \( \frac{\sqrt{37}}{2} \).

### Final Answer
The correct option is (b) \( \frac{\sqrt{37}}{2} \).

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