Let X be the midpoint of the side AB of triangle ABC. And Y be the mid point of CX. If Z is the point where BY meets the side AC, then `AZ:ZC` is equal to

Options

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

Answer

To solve the problem, we need to find the ratio \( AZ:ZC \) where \( Z \) is the point where line \( BY \) intersects side \( AC \) of triangle \( ABC \). Let's break down the solution step by step.

### Step 1: Define the Points
Let the coordinates of the points be defined as follows:
- \( A(0, 0) \)
- \( B(a, b) \)
- \( C(c, 0) \)

### Step 2: Find the Midpoint \( X \) of \( AB \)
The midpoint \( X \) of side \( AB \) can be calculated using the midpoint formula:
\[ X = \left( \frac{0 + a}{2}, \frac{0 + b}{2} \right) = \left( \frac{a}{2}, \frac{b}{2} \right) \]

### Step 3: Find the Midpoint \( Y \) of \( CX \)
Next, we find the midpoint \( Y \) of segment \( CX \):
\[ Y = \left( \frac{c + \frac{a}{2}}{2}, \frac{0 + \frac{b}{2}}{2} \right) = \left( \frac{c + \frac{a}{2}}{2}, \frac{b}{4} \right) \]

### Step 4: Find the Equation of Line \( BY \)
To find the equation of line \( BY \), we need the coordinates of points \( B(a, b) \) and \( Y \left( \frac{c + \frac{a}{2}}{2}, \frac{b}{4} \right) \).

Using the two-point form of the line equation:
\[ y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1) \]
Substituting \( (x_1, y_1) = (a, b) \) and \( (x_2, y_2) = \left( \frac{c + \frac{a}{2}}{2}, \frac{b}{4} \right) \):
\[ y - b = \frac{\frac{b}{4} - b}{\frac{c + \frac{a}{2}}{2} - a}(x - a) \]

### Step 5: Find the Equation of Line \( AC \)
The equation of line \( AC \) can be derived from points \( A(0, 0) \) and \( C(c, 0) \):
The slope \( m \) of line \( AC \) is:
\[ m = \frac{0 - 0}{c - 0} = 0 \]
Thus, the equation of line \( AC \) is simply \( y = 0 \).

### Step 6: Find Intersection Point \( Z \)
To find point \( Z \), we need to set the equations of lines \( BY \) and \( AC \) equal to each other. Since \( AC \) is on the x-axis (i.e., \( y = 0 \)), we substitute \( y = 0 \) into the equation of line \( BY \) and solve for \( x \).

### Step 7: Calculate the Ratio \( AZ:ZC \)
Let \( Z \) divide \( AC \) in the ratio \( k:1 \). The coordinates of \( Z \) can be expressed as:
\[ Z = \left( \frac{k \cdot c + 0}{k + 1}, \frac{k \cdot 0 + 0}{k + 1} \right) = \left( \frac{k \cdot c}{k + 1}, 0 \right) \]

Now, we can find the ratio \( AZ:ZC \) using the coordinates of \( A \) and \( C \):
\[ AZ = Z_x - A_x = \frac{k \cdot c}{k + 1} - 0 = \frac{k \cdot c}{k + 1} \]
\[ ZC = C_x - Z_x = c - \frac{k \cdot c}{k + 1} = \frac{c(k + 1 - k)}{k + 1} = \frac{c}{k + 1} \]

Thus, the ratio \( AZ:ZC \) becomes:
\[ AZ:ZC = \frac{k \cdot c}{k + 1} : \frac{c}{k + 1} = k:1 \]

### Conclusion
From the calculations, we find that the ratio \( AZ:ZC = 2:1 \).