Q,R and S are the points on line joining the points `P(a,x)` and `T(b,y)` such that `PQ=QR=RS=ST` then `((5a+3b)/8, (5x+3y)/8)` is the mid point of

Options

A. PQ
B. QR
C. RS
D. ST

Explanation

To solve the problem step by step, we need to find the midpoint of the segments formed by the points P(a, x) and T(b, y) with points Q, R, and S equally spaced on the line segment PT.

### Step 1: Identify the coordinates of points P and T
- Let \( P = (a, x) \)
- Let \( T = (b, y) \)

### Step 2: Determine the coordinates of point R
Since \( PQ = QR = RS = ST \), point R is the midpoint of segment PT. We can find the coordinates of R 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 T:
\[ R = \left( \frac{a + b}{2}, \frac{x + y}{2} \right) \]

### Step 3: Determine the coordinates of point Q
Point Q is the midpoint of segment PR. Again, we apply the midpoint formula:
\[ Q = \left( \frac{a + \frac{a + b}{2}}{2}, \frac{x + \frac{x + y}{2}}{2} \right) \]
Calculating the x-coordinate:
\[ Q_x = \frac{a + \frac{a + b}{2}}{2} = \frac{2a + a + b}{4} = \frac{3a + b}{4} \]
Calculating the y-coordinate:
\[ Q_y = \frac{x + \frac{x + y}{2}}{2} = \frac{2x + x + y}{4} = \frac{3x + y}{4} \]
Thus, the coordinates of Q are:
\[ Q = \left( \frac{3a + b}{4}, \frac{3x + y}{4} \right) \]

### Step 4: Determine the coordinates of point S
Point S is the midpoint of segment RT. We apply the midpoint formula again:
\[ S = \left( \frac{\frac{a + b}{2} + b}{2}, \frac{\frac{x + y}{2} + y}{2} \right) \]
Calculating the x-coordinate:
\[ S_x = \frac{\frac{a + b}{2} + b}{2} = \frac{a + b + 2b}{4} = \frac{a + 3b}{4} \]
Calculating the y-coordinate:
\[ S_y = \frac{\frac{x + y}{2} + y}{2} = \frac{x + y + 2y}{4} = \frac{x + 3y}{4} \]
Thus, the coordinates of S are:
\[ S = \left( \frac{a + 3b}{4}, \frac{x + 3y}{4} \right) \]

### Step 5: Find the midpoint of segment QR
Now, we need to find the midpoint of segment QR:
\[ \text{Midpoint of } QR = \left( \frac{Q_x + R_x}{2}, \frac{Q_y + R_y}{2} \right) \]
Substituting the coordinates of Q and R:
\[ \text{Midpoint of } QR = \left( \frac{\frac{3a + b}{4} + \frac{a + b}{2}}{2}, \frac{\frac{3x + y}{4} + \frac{x + y}{2}}{2} \right) \]
Calculating the x-coordinate:
\[ = \frac{\frac{3a + b}{4} + \frac{2(a + b)}{4}}{2} = \frac{\frac{3a + b + 2a + 2b}{4}}{2} = \frac{5a + 3b}{8} \]
Calculating the y-coordinate:
\[ = \frac{\frac{3x + y}{4} + \frac{2(x + y)}{4}}{2} = \frac{\frac{3x + y + 2x + 2y}{4}}{2} = \frac{5x + 3y}{8} \]
Thus, the midpoint of QR is:
\[ \left( \frac{5a + 3b}{8}, \frac{5x + 3y}{8} \right) \]

### Conclusion
The coordinates \( \left( \frac{5a + 3b}{8}, \frac{5x + 3y}{8} \right) \) represent the midpoint of segment QR.