Question

The area of a parallelogram is 2, and the lengths of its sides are 3.1 and 1. Determine, to the nearest tenth of a degree, the measure of the obtuse angle of the parallelogram.

Answer 1

To find the measure of the obtuse angle of the parallelogram, we can use the Law of Cosines. By using the formula for the area of a parallelogram and the Law of Cosines, we can solve for the measure of the obtuse angle. The answer is approximately 41.2°.

Step-by-step explanation:

To find the measure of the obtuse angle of the parallelogram, we can use the Law of Cosines. Let's call the lengths of the sides of the parallelogram a and b, and the measure of the obtuse angle θ. We know that cos(θ) = (a² + b² - c²) / (2ab), where c is the length of the diagonal of the parallelogram. Since the lengths of the sides are given as 3.1 and 1, and the area of the parallelogram is 2, we can use the formula for the area of a parallelogram, which is given by A = ab sin(θ), to find the length of the diagonal c.

Let's solve for c:

A = ab sin(θ)

2 = (3.1)(1) sin(θ)

sin(θ) = 2 / 3.1

θ ≈ arcsin(2/3.1) = 41.2°

Now we can use the Law of Cosines to find the measure of the obtuse angle:

cos(θ) = (a² + b² - c²) / (2ab)

cos(θ) = (3.1² + 1² - c²) / (2(3.1)(1))

cos(θ) = (9.61 + 1 - c²) / 6.2

c² = 10.61 - 6.2cos(θ)

c² ≈ 10.61 - 6.2cos(41.2°)

c ≈ √(10.61 - 6.2cos(41.2°))

c ≈ √(10.61 - 6.2(0.753))

c ≈ √(10.61 - 4.626)

c ≈ √6.984

c ≈ 2.64

Therefore, the measure of the obtuse angle of the parallelogram is approximately 41.2°.