Draw a line segment of length `sqrt(3)` cm.

Answer

To draw a line segment of length \( \sqrt{3} \) cm, we can use the method of constructing a right-angled triangle. Here’s a step-by-step solution:

### Step 1: Understand the Length
First, we need to express \( \sqrt{3} \) in a form that we can construct geometrically. We can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

### Step 2: Set Up the Right-Angled Triangle
We can express \( \sqrt{3} \) as \( \sqrt{2^2 - 1^2} \). This means we can create a right-angled triangle where one side is 1 cm and the other side is 2 cm.

### Step 3: Draw the Triangle
1. **Draw a line segment OA** of length 2 cm. This will be one side of the triangle.
2. **At point A**, draw a perpendicular line segment AB of length 1 cm. This will be the other side of the triangle.

### Step 4: Label the Points
Label the points as follows:
- O is the point where the 2 cm line starts.
- A is the endpoint of the 2 cm line.
- B is the endpoint of the 1 cm perpendicular line.

### Step 5: Use the Pythagorean Theorem
According to the Pythagorean theorem:
\[ OB^2 = OA^2 + AB^2 \]
Substituting the lengths we have:
\[ OB^2 = 2^2 + 1^2 \]
\[ OB^2 = 4 + 1 = 5 \]
Thus,
\[ OB = \sqrt{5} \]

### Step 6: Find the Length of OA
To find \( OA \):
\[ OA = \sqrt{OB^2 - AB^2} = \sqrt{4 - 1} = \sqrt{3} \]

### Step 7: Draw the Line Segment
Finally, draw the line segment from point O to point A, which will measure \( \sqrt{3} \) cm.

### Summary
You have now constructed a line segment of length \( \sqrt{3} \) cm using a right-angled triangle.

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