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To solve for the currents \( i_0 \), \( i_1 \), and \( i_2 \) in the given circuit, we will apply **Kirchhoff’s Voltage Law (KVL)** and **Ohm's Law**.

### Step 1: Solving for \( v_{\Delta} \)

Let's start with the left side of the circuit. 

- We have a voltage source of 18 V.
- Resistors of 12 Ω and 6 Ω are in series with each other.
  
The total resistance of the series combination is:
\[
R_{total} = 12 \, \Omega + 6 \, \Omega = 18 \, \Omega
\]

Using Ohm’s law, the current \( i_0 \) in this part of the circuit is:
\[
i_0 = \frac{18 \, V}{18 \, \Omega} = 1 \, A
\]

Now, we can calculate the voltage across the 6 Ω resistor, which is \( v_{\Delta} \):
\[
v_{\Delta} = i_0 \cdot 6 \, \Omega = 1 \, A \cdot 6 \, \Omega = 6 \, V
\]

### Step 2: Voltage at the parallel branch

Since the parallel part of the circuit has a controlled voltage source of \( v_{\Delta}/2 \), the voltage across the parallel branch is:
\[
v_{\Delta}/2 = \frac{6 \, V}{2} = 3 \, V
\]

This 3 V will be the same across both the 10 Ω and 5 Ω resistors in parallel.

### Step 3: Solving for \( i_1 \) and \( i_2 \)

Using Ohm’s Law, we can calculate the currents \( i_1 \) and \( i_2 \):

1. For the 10 Ω resistor (\( i_1 \)):
\[
i_1 = \frac{v_{\Delta}/2}{10 \, \Omega} = \frac{3 \, V}{10 \, \Omega} = 0.3 \, A
\]

2. For the 5 Ω resistor (\( i_2 \)):
\[
i_2 = \frac{v_{\Delta}/2}{5 \, \Omega} = \frac{3 \, V}{5 \, \Omega} = 0.6 \, A
\]

### Step 4: Checking the total current

Since \( i_1 \) and \( i_2 \) are the currents in parallel branches, the total current entering this parallel combination must be:
\[
i_{parallel} = i_1 + i_2 = 0.3 \, A + 0.6 \, A = 0.9 \, A
\]

This current \( i_{parallel} \) is equal to \( i_0 \) because of Kirchhoff’s Current Law (KCL). Therefore, the current through the 12 Ω resistor on the left is also 0.9 A.

### Summary of Results:
- \( i_0 = 1 \, A \)
- \( i_1 = 0.3 \, A \)
- \( i_2 = 0.6 \, A \)

Let me know if you want any more details or have questions!

#### 5 Related Questions:
1. What is Kirchhoff's Voltage Law and how is it applied in complex circuits?
2. How do you calculate equivalent resistance in parallel and series combinations?
3. Why does the voltage across parallel branches remain the same?
4. Can you explain the role of current division in parallel circuits?
5. How can we use Kirchhoff’s Current Law to verify circuit solutions?

#### Tip:
When solving for currents in a circuit, always apply Kirchhoff's Current and Voltage Laws systematically to simplify complex networks step by step.

This problem demonstrates how to use Kirchhoff’s Voltage and Current Laws along with Ohm’s Law to find the currents i0, i1, and i2 in a circuit. The solution includes calculating the total resistance, voltage across components, and applying these principles to solve for the unknowns.

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