Introduction

When you have the lengths of a parallelogram's sides and one of its diagonals, finding the area of the shape may seem challenging. However, with the right formula and a bit of trigonometry, it is absolutely doable. This guide will walk you through the process using the cosine rule, and it will also touch on Heron's formula for comparison.

Understanding the Problem

Given the parallelogram with sides a 100 cm, b 60 cm, and one diagonal d 80 cm, we aim to find the area of the parallelogram.

Step-by-Step Solution Using the Cosine Rule

Step 1: Applying the Cosine Rule

The Cosine Rule states: d^2 a^2 b^2 - 2ab cdot cosTheta.

Plugging in the values, we have:

80^2 100^2 60^2 - 2 cdot 100 cdot 60 cdot cosTheta

(Rightarrow 6400 10000 3600 - 12000 cdot cosTheta)

(Rightarrow 6400 13600 - 12000 cdot cosTheta)

(Rightarrow 12000 cdot cosTheta 7200)

(Rightarrow cosTheta 0.6)

Step 2: Finding (sinTheta)

Using the Pythagorean Identity: (1 - cos^2Theta sin^2Theta)

(Rightarrow sin^2Theta 1 - 0.6^2 1 - 0.36 0.64)

(Rightarrow sinTheta sqrt{0.64} 0.8)

Step 3: Calculating the Area

The area (A) of the parallelogram is given by the formula: (A a cdot b cdot sinTheta)

(Rightarrow A 100 cdot 60 cdot 0.8 4800 , text{cm}^2)

The final answer is the area of the parallelogram: 4800 cm2.

Using Heron's Formula for Comparison

Now, let's consider the alternative approach with Heron's formula. However, according to the correct formulation, 100 cm should be the diagonal. If the diagonal is 100 cm, the process is as follows:

1. Split the parallelogram into two right-angled triangles.

2. Let the angle between the sides 60 cm and 80 cm be 90°.

3. Use Heron's formula to find the area of each right-angled triangle and then sum them up.

Area of one triangle (frac{1}{2} cdot 60 cdot 80 2400 , text{cm}^2)

Thus, the area of the parallelogram (2 cdot 2400 4800 , text{cm}^2)

Therefore, the correct area is again 4800 cm2 when using the correct diagonal length.

Conclusion

Regardless of the method used, the area of the parallelogram with the given sides and one diagonal is 4800 cm2.

Key Takeaways

Use the Cosine Rule to find the angle between the sides and the diagonal. Apply the sine value to the formula: (A a cdot b cdot sinTheta). Use Heron's formula for comparison in triangulated cases.