Final answer:
To construct a line segment with length √2, draw a line segment of length 1, erect a perpendicular line segment of the same length at one end, and the hypotenuse of the resulting right-angled triangle will be the desired length √2.
Step-by-step explanation:
Constructing a Line Segment with Length √2
To construct a line segment whose length is √2, we can utilize the Pythagorean theorem by creating a right-angle triangle with two sides of length 1. Since we are given a line segment of length 1, this will be our first side. The triangle will then have sides of length 1, 1, and the hypotenuse will be the length we desire, √2, according to the relation given by the Pythagorean theorem: a² + b² = c².
- Draw a line segment of length 1 (this will be one of the sides of the right-angled triangle).
- At one end of this segment, construct a perpendicular line segment, also of length 1.
- Connect the ends of these two line segments. The length of this newly formed line segment will be √2, as it is the hypotenuse of a right-angle triangle with sides of equal length.