P and Q are points on the line joining A(-2,5) and B(3,1) such that AP=PQ=QB. Then the mid pont of PQ is

Options

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

Answer

To find the midpoint of points P and Q on the line segment joining A(-2, 5) and B(3, 1) such that AP = PQ = QB, we can follow these steps:

### Step 1: Determine the Coordinates of Points A and B
We have:
- A = (-2, 5)
- B = (3, 1)

### Step 2: Calculate the Distance AB
To find the distance between points A and B, we can use the distance formula:
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of A and B:
\[ AB = \sqrt{(3 - (-2))^2 + (1 - 5)^2} = \sqrt{(3 + 2)^2 + (1 - 5)^2} = \sqrt{5^2 + (-4)^2} = \sqrt{25 + 16} = \sqrt{41} \]

### Step 3: Determine the Length of AP, PQ, and QB
Since AP = PQ = QB, we can denote the length of each segment as \(d\). Since the total length AB is divided into three equal parts:
\[ AP + PQ + QB = AB \implies 3d = \sqrt{41} \implies d = \frac{\sqrt{41}}{3} \]

### Step 4: Find the Coordinates of Point P
Point P divides the segment AB in the ratio 1:2. We can use the section formula:
\[ P\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]
where \(m = 1\), \(n = 2\), \(A(-2, 5)\) and \(B(3, 1)\):
\[ P\left(\frac{1 \cdot 3 + 2 \cdot (-2)}{1 + 2}, \frac{1 \cdot 1 + 2 \cdot 5}{1 + 2}\right) = P\left(\frac{3 - 4}{3}, \frac{1 + 10}{3}\right) = P\left(\frac{-1}{3}, \frac{11}{3}\right) \]

### Step 5: Find the Coordinates of Point Q
Point Q divides the segment AB in the ratio 2:1. Again using the section formula:
\[ Q\left(\frac{2 \cdot 3 + 1 \cdot (-2)}{2 + 1}, \frac{2 \cdot 1 + 1 \cdot 5}{2 + 1}\right) = Q\left(\frac{6 - 2}{3}, \frac{2 + 5}{3}\right) = Q\left(\frac{4}{3}, \frac{7}{3}\right) \]

### Step 6: Find the Midpoint of PQ
The midpoint R of segment PQ can be found using the midpoint formula:
\[ R\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]
Substituting the coordinates of P and Q:
\[ R\left(\frac{\frac{-1}{3} + \frac{4}{3}}{2}, \frac{\frac{11}{3} + \frac{7}{3}}{2}\right) = R\left(\frac{3/3}{2}, \frac{18/3}{2}\right) = R\left(\frac{1}{2}, 3\right) \]

### Final Answer
The midpoint of PQ is \(\left(\frac{1}{2}, 3\right)\).