Calculating the Perimeter of a Parallelogram Using the Intersection of Diagonals and Area

Parallelograms are intriguing geometric shapes due to their symmetrical properties. Understanding these properties can help us calculate various aspects, such as the perimeter, given certain measurements involving the intersection of their diagonals and their area. In this article, we will explore how to calculate the perimeter of a parallelogram using the given area and the distances from the intersection of diagonals to the sides.

The Area and Intersection of Diagonals

Consider a parallelogram with an area of 30 cm2. The point of intersection of the diagonals, known as the midpoint, divides the parallelogram into four smaller triangles, with opposite triangles being congruent and having equal areas. Since the total area of the parallelogram is 30 cm2, it means that each of these four triangles has an area of:

15 cm2

Determining the Heights and Bases of the Triangles

The problem states that the intersection point of the diagonals is 3 cm and 1.5 cm from the sides. This information can be used to determine the heights of the triangles. Let's denote the heights as follows:

h1 3 cm h2 1.5 cm

Given these heights, we can calculate the bases of the triangles:

For the triangles with height h1 (3 cm), the base is calculated as:

15 × 2 / 3 10 cm

For the triangles with height h2 (1.5 cm), the base is calculated as:

15 × 2 / 2.5 12 cm

Since the parallelogram is symmetrical, these bases represent the lengths of the opposite sides.

Calculating the Perimeter

Now that we have the bases of the triangles, we can calculate the perimeter of the parallelogram. We know that the opposite sides are equal in length, so the perimeter will be:

2 (Base1 Base2) 2 (10 12) 44 cm

Revisiting the Example Problem

Let's re-examine a similar problem to understand the application of these principles:

Given a parallelogram ABCD with an area of 30 square cm, the diagonals AC and BD intersect at point O. Perpendiculars are drawn from point O to BC and AB as OP and OQ, respectively. OQ 3 cm and OP 1.5 cm. Extend PO and QO, and they intersect AD and CD at R and S, respectively. Using the congruence of triangles, we can deduce:

OR OP 3 cm (since triangles OAR and OCP are congruent) QS 2 × OQ 6 cm (since triangles OAQ and OCS are congruent)

Given these values, we can determine the lengths of the sides:

BC 10 cm (since BC × PR 30 cm2, and PR 3 cm) AB 5 cm (since AB × QS 30 cm2, and QS 6 cm)

Thus, the opposite sides are equal, so:

AD BC 10 cm and DC AB 5 cm

Therefore, the perimeter of the parallelogram ABCD is:

2 (10 5) 30 cm

Conclusion

The perimeter calculation of a parallelogram can be effectively derived using the intersection of its diagonals and the area, as demonstrated in the examples above. Understanding these principles not only aids in problem-solving but also reinforces the geometric properties of parallelograms. The key steps involve recognizing the congruence of the smaller triangles formed by the intersection and applying the area formulas accordingly.