Representing [tex]\(\sqrt{2}\)[/tex] on the number line involves a geometric construction. Here, I will provide a step-by-step explanation of how to achieve this:
### Step-by-Step Construction
1. Draw a Number Line:
- Start by drawing a horizontal line and mark a point '0' on it to represent zero.
- Mark another point '1' to the right of '0' to represent the number one. Ensure that the distance between '0' and '1' is equal to one unit.
2. Construct a Perpendicular Line:
- At point '1', draw a vertical line perpendicular to the number line.
- Mark a point 'A' on this perpendicular line such that the distance from '1' to 'A' is also one unit.
3. Forming a Right Triangle:
- Connect point '0' to point 'A' with a straight line. This line segment is the hypotenuse of a right triangle with legs of length 1 unit each.
4. Finding the Hypotenuse ([tex]\(\sqrt{2}\)[/tex]):
- By the Pythagorean theorem, the length of the hypotenuse of this right triangle is [tex]\(\sqrt{1^2 + 1^2} = \sqrt{1 + 1} = \sqrt{2}\)[/tex].
5. Transferring [tex]\(\sqrt{2}\)[/tex] to the Number Line:
- Use a compass to measure the length of the hypotenuse '0A' (which is [tex]\(\sqrt{2}\)[/tex]).
- Place the compass point at '0' and draw an arc that intersects the number line. Mark this intersection point as [tex]\( \sqrt{2} \)[/tex].
Now, the length from '0' to this new point represents [tex]\(\sqrt{2}\)[/tex] on the number line.
### Summary of Key Points:
- You start by marking '0' and '1' on the number line.
- Draw a perpendicular from '1' and mark a point one unit above '1'.
- Form a right triangle and use the Pythagorean theorem to understand the hypotenuse length is [tex]\(\sqrt{2}\)[/tex].
- Transfer this hypotenuse length to the number line as the representation of [tex]\(\sqrt{2}\)[/tex].
This geometric method provides a clear visual representation of [tex]\(\sqrt{2}\)[/tex] on the number line.
