If `z_(1)=5+7i`, then `|-z_(1)|=|z_(1)|`. is `True `

Answer

To prove that \( |-z_1| = |z_1| \) for \( z_1 = 5 + 7i \), we will follow these steps:

### Step 1: Calculate the modulus of \( z_1 \)
The modulus of a complex number \( z = a + bi \) is given by the formula:
\[ |z| = \sqrt{a^2 + b^2} \]
For \( z_1 = 5 + 7i \):
- Here, \( a = 5 \) and \( b = 7 \).

Now, we calculate:
\[ |z_1| = \sqrt{5^2 + 7^2} = \sqrt{25 + 49} = \sqrt{74} \]

### Step 2: Calculate the modulus of \( -z_1 \)
Now, we need to find \( -z_1 \):
\[ -z_1 = - (5 + 7i) = -5 - 7i \]

Next, we calculate the modulus of \( -z_1 \):
\[ |-z_1| = |-5 - 7i| = \sqrt{(-5)^2 + (-7)^2} = \sqrt{25 + 49} = \sqrt{74} \]

### Step 3: Compare the moduli
Now we compare the two moduli:
\[ |-z_1| = \sqrt{74} \quad \text{and} \quad |z_1| = \sqrt{74} \]

Since both moduli are equal, we conclude that:
\[ |-z_1| = |z_1| \]

### Conclusion
Thus, the statement \( |-z_1| = |z_1| \) is **True**.

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