Given segment AB, explain how to construct a square with sides of length AB.
The Correct Answer and Explanation is:
To construct a square with sides equal to the length of segment AB, follow these steps:
Construction Steps:
- Draw segment AB using a straightedge. This will be one side of the square.
- Construct a perpendicular line at point A using a compass and straightedge (perpendicular to segment AB).
- Copy the length AB onto the perpendicular line starting at point A. Label this new point D, so that AD = AB.
- Construct a line segment parallel to AB through point D.
- Construct a line segment parallel to AD through point B.
- The intersection of the lines from steps 4 and 5 is point C.
- Connect the points to form the square: A to B, B to C, C to D, and D to A.
Explanation
Constructing a square from a given line segment involves both geometric precision and an understanding of perpendicular and parallel lines. The defining feature of a square is that all four sides are of equal length, and all angles are 90 degrees. Given segment AB, we use it as one side of the square.
The first step is to draw segment AB, which sets the base. To construct a square, we need a right angle from one endpoint (say point A), which can be achieved using a compass. By constructing a perpendicular at point A and measuring the same length as AB on this line, we get point D such that AD = AB and ∠DAB = 90°, forming the adjacent side.
To complete the square, we must draw lines from points D and B that are parallel to AB and AD respectively. These lines intersect at point C. By construction, the new segments BC and CD will also be equal in length to AB because they are copied from AB and created through right-angle constructions.
This method ensures that the resulting quadrilateral ABCD has four equal sides and four right angles, which meets the definition of a square. This construction only uses a compass and straightedge, adhering to classical Euclidean geometry principles.
This geometric process is not only mathematically rigorous but also foundational for understanding symmetry, transformations, and the basic structure of geometric figures. It reinforces spatial reasoning and the importance of congruence and angle construction in geometry.
