if `z_(1)=3+4i and z_(2)=12-5i`, verify:
(i) `|-z_(1)|=|z_(1)|`
(ii) `|z_(1)+z_(2)|lt|z_(1)|+|z_(2)|`
(iii) ` |z_(1)z_(2)|=|z_(1)||z_(2)|`.

Answer

To verify the given statements about the complex numbers \( z_1 = 3 + 4i \) and \( z_2 = 12 - 5i \), we will go through each part step by step.

### Part (i): Verify \( |-z_1| = |z_1| \)

1. **Calculate \( |z_1| \)**:
\[ |z_1| = |3 + 4i| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

2. **Calculate \( |-z_1| \)**:
\[ |-z_1| = |- (3 + 4i)| = |-3 - 4i| = \sqrt{(-3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

3. **Conclusion**:
\[ |-z_1| = |z_1| \implies 5 = 5 \]
Thus, the statement is verified.

### Part (ii): Verify \( |z_1 + z_2| < |z_1| + |z_2| \)

1. **Calculate \( z_1 + z_2 \)**:
\[ z_1 + z_2 = (3 + 4i) + (12 - 5i) = 3 + 12 + (4 - 5)i = 15 - i \]

2. **Calculate \( |z_1 + z_2| \)**:
\[ |z_1 + z_2| = |15 - i| = \sqrt{15^2 + (-1)^2} = \sqrt{225 + 1} = \sqrt{226} \]

3. **Calculate \( |z_2| \)**:
\[ |z_2| = |12 - 5i| = \sqrt{12^2 + (-5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \]

4. **Calculate \( |z_1| + |z_2| \)**:
\[ |z_1| + |z_2| = 5 + 13 = 18 \]

5. **Compare \( |z_1 + z_2| \) and \( |z_1| + |z_2| \)**:
\[ \sqrt{226} \approx 15.1 < 18 \]
Thus, \( |z_1 + z_2| < |z_1| + |z_2| \) is verified.

### Part (iii): Verify \( |z_1 z_2| = |z_1| |z_2| \)

1. **Calculate \( z_1 z_2 \)**:
\[ z_1 z_2 = (3 + 4i)(12 - 5i) = 3 \cdot 12 + 3 \cdot (-5i) + 4i \cdot 12 + 4i \cdot (-5i) \]
\[ = 36 - 15i + 48i - 20i^2 \]
Since \( i^2 = -1 \):
\[ = 36 - 15i + 48i + 20 = 56 + 33i \]

2. **Calculate \( |z_1 z_2| \)**:
\[ |z_1 z_2| = |56 + 33i| = \sqrt{56^2 + 33^2} = \sqrt{3136 + 1089} = \sqrt{4225} = 65 \]

3. **Calculate \( |z_1| |z_2| \)**:
\[ |z_1| |z_2| = 5 \cdot 13 = 65 \]

4. **Conclusion**:
\[ |z_1 z_2| = |z_1| |z_2| \implies 65 = 65 \]
Thus, the statement is verified.

### Summary of Results
- Part (i): \( |-z_1| = |z_1| \) is verified.
- Part (ii): \( |z_1 + z_2| < |z_1| + |z_2| \) is verified.
- Part (iii): \( |z_1 z_2| = |z_1| |z_2| \) is verified.