How to Construct a Parallelogram with Given Side Lengths and Diagonal

The construction of a parallelogram with given side lengths and diagonal involves understanding the properties of parallelograms and using geometric principles to ensure accuracy. This article will guide you through the process of constructing a parallelogram ABCD with AB 5 cm, BC 3 cm, and diagonal AC 6 cm. We will also explore a second method for the construction of the same parallelogram.

Mathematical Analysis

First, let's analyze the given values using the Pythagorean theorem. For a parallelogram, the diagonals bisect each other. If AC is the shorter diagonal, then the other diagonal BD would be longer. We can check the validity of AC being the shorter diagonal by comparing the sum of the squares of the sides to the square of the diagonal.

Given sides: AB 5 cm, BC 3 cm, and diagonal AC 6 cm.

Let's verify:

AB2 BC2 52 32 25 9 34

AC2 62 36

Since 34

Geometric Construction Using a Compass

Method 1: Using the Shorter Diagonal

To construct the parallelogram, follow these steps:

Draw a line segment AB of length 5 cm.With A as the center and a radius of 5 cm, draw an arc.With B as the center and a radius of 3 cm, draw an arc that intersects the previous arc at point C.With A as the center and a radius of 3 cm, draw an arc on the other side of AC.With C as the center and a radius of 5 cm, draw an arc that intersects the previous arc at point AD, DC, and CB to complete the parallelogram ABCD.

Method 2: Using Symmetry and Geometry

To construct the parallelogram using a different method:

Draw a line segment AB of 5 cm.With A as the center and a radius of 3 cm, draw arcs on both sides.With B as the center and a radius of 3 cm, draw arcs that intersect the previous arcs at points C and D.With B as the center and a radius of 6 cm, draw an arc to cut the arc centered at A at point D.With A as the center and a radius of 6 cm, draw an arc to cut the arc centered at B at point AD, DC, and CB to form the parallelogram ABCD.

Conclusion

Constructing a parallelogram with given side lengths and diagonal involves precise measurements and careful drawing. Both methods described above ensure that the geometric properties of a parallelogram are preserved. Understanding these methods can be beneficial for students and professionals working in fields that require geometric constructions, such as architecture, engineering, and design.