Use the golden ratio to divide the line segment L with length of 12 units into two segments L, and L. What are the lengths rounded off to two decimal places?

To divide the line segment \( L \) of length 12 units into two segments using the golden ratio, we use the following steps:

### Step 1: Understanding the Golden Ratio
The golden ratio, denoted by \( \phi \), is approximately \( 1.618 \). When a line segment is divided according to the golden ratio, the ratio of the whole line \( L \) to the longer segment \( x \) is equal to the ratio of \( x \) to the shorter segment \( y \):

\[
\frac{L}{x} = \frac{x}{y} = \phi
\]

### Step 2: Express the lengths in terms of the golden ratio
Let the length of the longer segment be \( x \) and the shorter segment be \( y \). We have:

\[
L = x + y
\]

And according to the golden ratio:

\[
\frac{x}{y} = \phi
\]

So:

\[
x = \phi y
\]

Substitute this into the equation \( L = x + y \):

\[
12 = \phi y + y
\]

### Step 3: Solve for \( y \)
Factor out \( y \):

\[
12 = y(\phi + 1)
\]

Thus:

\[
y = \frac{12}{\phi + 1}
\]

### Step 4: Solve for \( x \)
Since \( x = \phi y \):

\[
x = \phi \cdot \frac{12}{\phi + 1}
\]

### Step 5: Substitute the value of \( \phi \)
\(\phi = \frac{1 + \sqrt{5}}{2} \approx 1.618\), and \( \phi + 1 \approx 2.618 \):

\[
y = \frac{12}{2.618} \approx 4.58 \text{ units}
\]

\[
x = 1.618 \times 4.58 \approx 7.42 \text{ units}
\]

### Final Answer:
- The longer segment \( x \) is approximately **7.42** units.
- The shorter segment \( y \) is approximately **4.58** units.

Would you like further details or have any questions?

Here are some related questions you might find interesting:

1. What is the exact value of the golden ratio \( \phi \)?
2. How is the golden ratio related to the Fibonacci sequence?
3. Can you divide any arbitrary length using the golden ratio?
4. What is the significance of the golden ratio in geometry and art?
5. How can you derive the value of the golden ratio algebraically?

**Tip:** The golden ratio is often found in nature and art, creating aesthetically pleasing proportions.

Learn how to divide a line segment of length 12 units using the golden ratio. Calculate the lengths of the segments L and L using step-by-step mathematical calculations. Includes detailed explanations and rounded values to two decimal places.

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