D is any point on side AC of `DeltaABC`. If P,Q, X, Y are the mid points of `AB, BC, AD and DC `respectively, then the ratio of `PX and QY `is

Options

A. `1 : 2 `
B. `1:1`
C. `2:1`
D. `2:3 `

Answer

To solve the problem, we need to find the ratio of the lengths \( PX \) and \( QY \) where \( P, Q, X, Y \) are the midpoints of the segments \( AB, BC, AD, \) and \( DC \) respectively in triangle \( ABC \) with point \( D \) on side \( AC \).

### Step-by-step Solution:

1. **Identify the Points and Midpoints**:
- Let \( A \), \( B \), and \( C \) be the vertices of triangle \( ABC \).
- Let \( D \) be any point on side \( AC \).
- Define the midpoints:
- \( P \) is the midpoint of \( AB \).
- \( Q \) is the midpoint of \( BC \).
- \( X \) is the midpoint of \( AD \).
- \( Y \) is the midpoint of \( DC \).

2. **Use Coordinate Geometry**:
- Assign coordinates to the points:
- Let \( A(0, 0) \), \( B(2b, 0) \), and \( C(2c, 2h) \).
- The coordinates of point \( D \) on line segment \( AC \) can be represented as \( D(2k, 2m) \) where \( k \) and \( m \) are parameters that define the position of \( D \) on \( AC \).

3. **Calculate the Midpoints**:
- The coordinates of the midpoints can be calculated as follows:
- \( P = \left( \frac{0 + 2b}{2}, \frac{0 + 0}{2} \right) = (b, 0) \)
- \( Q = \left( \frac{2b + 2c}{2}, \frac{0 + 2h}{2} \right) = (b + c, h) \)
- \( X = \left( \frac{0 + 2k}{2}, \frac{0 + 2m}{2} \right) = (k, m) \)
- \( Y = \left( \frac{2k + 2c}{2}, \frac{2m + 2h}{2} \right) = (k + c, m + h) \)

4. **Find the Lengths \( PX \) and \( QY \)**:
- Calculate the distance \( PX \):
\[ PX = \sqrt{(k - b)^2 + (m - 0)^2} \]
- Calculate the distance \( QY \):
\[ QY = \sqrt{((k + c) - (b + c))^2 + ((m + h) - h)^2} = \sqrt{(k - b)^2 + m^2} \]

5. **Establish the Ratio \( \frac{PX}{QY} \)**:
- From the calculations, we see that:
\[ PX = \sqrt{(k - b)^2 + m^2} \]
\[ QY = \sqrt{(k - b)^2 + m^2} \]
- Therefore, the ratio \( \frac{PX}{QY} \) simplifies to:
\[ \frac{PX}{QY} = \frac{\sqrt{(k - b)^2 + m^2}}{\sqrt{(k - b)^2 + m^2}} = 1 \]

### Final Answer:
The ratio of \( PX \) to \( QY \) is \( 1:1 \).