Final answer:
To represent √19 on a number line, draw a number line, mark point 4 (the closest perfect square to 19), create a perpendicular line from this point with length 3, resulting in a right triangle. By the Pythagorean theorem, the hypotenuse represents √19, which you can then mark on the line.
Step-by-step explanation:
To represent √19 on a number line, you can utilize the concept of square roots as fractional powers and the Pythagorean theorem. Follow these steps:
- Draw a number line and mark a point O for the origin (0) and a point A where 4 is located since 4 is the perfect square closest to 19 without exceeding it.
- From point A, draw a line segment perpendicular to the number line. The length of this line segment should be equal to 3, since 3 when squared (3²) plus 4 squared (4²) will give you 19 (√19).
- Now you have a right triangle OAB, where OA = 4 and AB = 3. Using the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides, we verify that the hypotenuse OB represents √19. That is, OB = √(OA² + AB²) = √19.
- Mark the point B on the number line as √19.
Through this geometric process, you have precisely represented the square root of 19 on a number line.