To represent $\sqrt{5}$ on a number line, follow these steps:

1. Draw the Number Line: Start by drawing a straight horizontal line. Mark the point $0$ (origin) and then mark points for $1$, $2$, $3$, etc., equally spaced on the right side of the line.

2. Mark a Unit Segment: From $0$, measure a segment that corresponds to 2 units on the number line. So you should have the points $0$ and $2$ marked clearly.

3. Construct a Right Triangle: At the point $2$, draw a perpendicular line (a vertical line). From $2$, measure a height of exactly 1 unit up along the vertical line. This creates a right triangle with one leg along the number line of length 2 units and another leg perpendicular to it of length 1 unit.

4. Find the Hypotenuse: The length of the hypotenuse of this triangle will be $\sqrt{(2^2 + 1^2)} = \sqrt{5}$. This is the length we are interested in.

5. Plot $\sqrt{5}$ on the Number Line: Using a compass, set the radius of the compass to the length of the hypotenuse (i.e., $\sqrt{5}$) and draw an arc with the center at point $0$ on the number line. The point where this arc intersects the number line is the position of $\sqrt{5}$.

This method shows how to geometrically construct $\sqrt{5}$ on a number line using a right triangle and the Pythagorean theorem.

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Would you like further details or clarification? Here are some follow-up questions:

1. How would you represent $\sqrt{2}$ on the number line?
2. Can you explain why $\sqrt{5}$ is irrational?
3. What is the significance of the Pythagorean theorem in this construction?
4. How would you modify this construction for other square roots like $\sqrt{10}$?
5. What is the geometric interpretation of square roots?

Tip: The Pythagorean theorem is a powerful tool for representing square roots on a number line geometrically.