Question

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

Answer 1

We will represent \( \sqrt{5} \) on the number line using geometric construction. Here are the steps:

Step-by-Step Construction:

1. Draw a Line Segment of Length 5:

Start by drawing a line segment of length 5 units along the number line, with the starting point as 0. This represents the number 5.

2. Construct a Perpendicular Line from the Endpoint of the Line Segment:

At the point where the line segment ends (i.e., at the point representing 5), construct a perpendicular line segment of length 4 units. This step is based on the idea that \( \sqrt{5} \) can be represented by the diagonal of a right-angled triangle whose sides are of lengths 2 and 1.

3. Use the Pythagorean Theorem:

The hypotenuse of the right triangle formed will represent \( \sqrt{5} \), based on the Pythagorean theorem. In this case, the length of the hypotenuse will be the square root of the sum of the squares of the other two sides. So the hypotenuse of a triangle with legs of length 2 and 1 will have a length of \( \sqrt{5} \).

4. Mark the Point on the Number Line:

The point on the number line that corresponds to \( \sqrt{5} \) will be at a distance that matches the length of the hypotenuse in the constructed triangle.

Final Representation:

The value of \( \sqrt{5} \) is approximately 2.236, so on a number line, it will be placed slightly to the right of 2 and less than 3.

Final Answer:

\[ \sqrt{5} \approx 2.236 \] This point can be represented geometrically on the number line by following the above construction steps.