X, Y and Z are points on the sides AB, BC and AC of the triangle ABC, such that AX: XB = 4:3, BY: YC=2:3 and CZ: ZA=2:1. Find the ratio of the area of the triangle XYZ to that of the triangle ABC

To solve this problem, we will use **Mass Point Geometry**, a powerful method to handle ratios of areas and lines in geometric figures, particularly in triangles.

### Step 1: Assigning masses to the points
We begin by assigning masses to points based on the given ratios of line segments:

1. **AX : XB = 4 : 3**: Assign mass \( m_A = 3 \) and \( m_B = 4 \) because the ratio is the reciprocal of the segment division. This ensures that the point \( X \) divides \( AB \) in a 4:3 ratio.
   
2. **BY : YC = 2 : 3**: Similarly, assign mass \( m_B = 3 \) and \( m_C = 2 \) so that point \( Y \) divides \( BC \) in a 2:3 ratio.

3. **CZ : ZA = 2 : 1**: Lastly, assign mass \( m_C = 1 \) and \( m_A = 2 \) so that point \( Z \) divides \( AC \) in a 2:1 ratio.

### Step 2: Adjusting the masses
Now, we need to adjust the masses so that they are consistent across all points. We already have three conflicting masses for point \( B \) and point \( C \), so we must adjust them to a common set of values:

- From the \( AX:XB = 4:3 \) relation, we have \( m_A = 3 \) and \( m_B = 4 \).
- From the \( BY:YC = 2:3 \) relation, we had set \( m_B = 3 \) and \( m_C = 2 \), but we need to ensure these masses align. To make this consistent, multiply all masses by the least common multiple of the masses of \( B \), which is 12.

Thus:
- \( m_A = 9 \)
- \( m_B = 12 \)
- \( m_C = 8 \)

### Step 3: Determining the area ratio
In mass point geometry, the area of triangle \( XYZ \) is related to the area of triangle \( ABC \) by the product of the masses at the vertices.

The area ratio of triangles \( XYZ \) and \( ABC \) is given by the formula:
\[
\text{Area ratio} = \frac{1}{(m_A + m_B + m_C)} = \frac{1}{9 + 12 + 8} = \frac{1}{29}
\]

Thus, the ratio of the area of triangle \( XYZ \) to the area of triangle \( ABC \) is:
\[
\boxed{\frac{1}{29}}
\]

This completes the solution.

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Would you like further clarification on any step?

### Follow-up Questions:
1. What is mass point geometry and how is it used in other triangle problems?
2. How do we derive the formula for area ratios using masses?
3. How does the mass point method differ from using coordinates in geometry?
4. Can this method be applied to non-triangular shapes, and how would that work?
5. What happens if we change the ratio of the segments?

### Tip:
Mass point geometry is especially useful when dealing with multiple points dividing sides of a triangle, and it greatly simplifies complex area and ratio problems.

X, Y and Z are points on the sides AB, BC and AC of the triangle ABC, such that AX: XB = 4:3, BY: YC = 2:3 and CZ: ZA = 2:1. Find the ratio of the area of the triangle XYZ to that of the triangle ABC.

Discover how to find the ratio of the area of triangle XYZ to triangle ABC using Mass Point Geometry. This problem involves dividing the sides of a triangle and calculating the area ratio with mass points. A perfect challenge for high school students in Grades 10-12.

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