If `z_(1),z_(2),z_(3)` and u,v,w are complex numbers represending the vertices of two triangles such that `z_(3)=(1-t)z_(1)+tz_(2) and w = (1 -u) u +tv`, where t is a complex number, then the two triangles

Options

A. Have the same area
B. Are similar
C. Are congruent
D. Are equilateral

Answer

To solve the problem, we need to analyze the given complex numbers representing the vertices of two triangles and show that they are similar.

### Step-by-Step Solution:

1. **Identify the vertices of the triangles:**
Let the vertices of the first triangle be represented by the complex numbers \( z_1, z_2, z_3 \) and the vertices of the second triangle be represented by \( u, v, w \).

2. **Given relationships:**
We have the relationships:
\[ z_3 = (1-t)z_1 + tz_2 \]
\[ w = (1-u)u + tv \]
Here, \( t \) is a complex number.

3. **Calculate the sides of the triangles:**
- For triangle \( ABC \) with vertices \( z_1, z_2, z_3 \):
- Side \( AB = z_2 - z_1 \)
- Side \( BC = z_3 - z_2 = [(1-t)z_1 + tz_2] - z_2 = (1-t)z_1 + (t-1)z_2 = (1-t)(z_1 - z_2) \)
- Side \( CA = z_1 - z_3 = z_1 - [(1-t)z_1 + tz_2] = z_1 - (1-t)z_1 - tz_2 = t(z_1 - z_2) \)

- For triangle \( PQR \) with vertices \( u, v, w \):
- Side \( PQ = v - u \)
- Side \( QR = w - v = [(1-u)u + tv] - v = (1-u)u + (t-1)v = (1-u)(u - v) \)
- Side \( RP = u - w = u - [(1-u)u + tv] = u - (1-u)u - tv = u - (1-u)u - tv = t(u - v) \)

4. **Establish ratios of corresponding sides:**
Now we can establish the ratios of the corresponding sides of the triangles:
- Ratio of sides \( AB \) and \( PQ \):
\[ \frac{AB}{PQ} = \frac{z_2 - z_1}{v - u} \]
- Ratio of sides \( BC \) and \( QR \):
\[ \frac{BC}{QR} = \frac{(1-t)(z_1 - z_2)}{(1-u)(u - v)} \]
- Ratio of sides \( CA \) and \( RP \):
\[ \frac{CA}{RP} = \frac{t(z_1 - z_2)}{t(u - v)} \]

5. **Simplifying the ratios:**
Since \( t \) is a common factor in the last ratio and can be canceled out, we can analyze the ratios:
- The ratios \( \frac{AB}{PQ} \), \( \frac{BC}{QR} \), and \( \frac{CA}{RP} \) are all equal, showing that the triangles are similar.

6. **Conclusion:**
Therefore, we conclude that triangle \( ABC \) is similar to triangle \( PQR \).