Calculating Parallelogram Area with Heights QM and QN

Study Notes

Discovering the Area of a Parallelogram with Heights QM and QN

Imagine a parallelogram named PQRS, where Q and R are the opposite vertices, and SR has a length of 12 centimeters. We'll also find the area of this parallelogram using the heights QM and QN (see Figure 1).

Parallelogram PQRS

Figure 1: Parallelogram PQRS with heights QM = 7.6 cm and QN

To calculate the area of a parallelogram, we can use either base and height or diagonal and base. In this case, we'll use the base PR (PS) and the height QM.

The area (A) of a parallelogram is given by:

[ A = b \cdot h ]

where ( b ) is the length of the base and ( h ) is the height.

In our case, the base PR (PS) is equal to the diagonal SR since PR and PS are both perpendicular to SR. We can find the length of SR using the Pythagorean theorem:

[ SR^2 = SP^2 + PR^2 ]

[ SR^2 = (12/2)^2 + (12/2)^2 ]

[ SR^2 = 36 + 36 = 72 ]

[ SR = \sqrt{72} = 6\sqrt{3} \text{ centimeters} ]

Since PR = SR, our base is 6√3 cm.

Now that we have the base (PR = PS), we need to find the height QM. By definition, heights of a parallelogram are perpendicular to the base and are congruent (equal). Therefore, QN = QM:

[ A = PR \cdot QM = (6\sqrt{3}) \cdot 7.6 ]

[ A = 45.6 \sqrt{3} \text{ square centimeters} ]

QED.

So the area of our parallelogram PQRS is approximately 45.6√3 square centimeters, which is approximately 67.9 square centimeters (rounded to two decimal places).

This principle is handy when you're dealing with parallelograms and looking to calculate their area without using more advanced trigonometry or geometry concepts.

Description

Learn how to find the area of a parallelogram using the heights QM and QN, based on the lengths of the base and diagonals. Explore the step-by-step process with measurements and the Pythagorean theorem to determine the area efficiently.