Show how √5 can be represented on the number line.

Sure, here are the steps to represent √5 on the number line:

1. Draw a straight horizontal line and mark a point on it as O. This will be our origin or zero point.

2. From point O, draw a line segment OA of unit length towards the right side. You can use a ruler to measure this unit length.

3. Now, draw another line segment AB of unit length from point A to the right side.

4. Now, we have two line segments OA and AB of unit length. We need to find a point that represents √5 on the number line.

5. To do this, draw a line segment OB. This will form a right-angled triangle OAB.

6. Now, using the Pythagorean theorem, we know that OB^2 = OA^2 + AB^2. Since OA and AB are of unit length, OB^2 = 1^2 + 1^2 = 2.

7. To get √5, we need to extend the line segment OB by a length of √3. To do this, draw a line segment BC of unit length at a right angle to OB at point B.

8. Now, draw the line segment OC. This will form another right-angled triangle OBC.

9. Using the Pythagorean theorem again, we know that OC^2 = OB^2 + BC^2. Since OB = √2 and BC = 1, OC^2 = (√2)^2 + 1^2 = 2 + 1 = 3.

10. To get √5, we need to extend the line segment OC by a length of √2. To do this, draw a line segment CD of unit length at a right angle to OC at point C.

11. Now, draw the line segment OD. This will form another right-angled triangle OCD.

12. Using the Pythagorean theorem again, we know that OD^2 = OC^2 + CD^2. Since OC = √3 and CD = 1, OD^2 = (√3)^2 + 1^2 = 3 + 1 = 4.

13. Therefore, the length of OD is √4 = 2. This means that the point D represents √5 on the number line.

14. Mark point D on the number line. This point represents √5 on the number line.