Problem 40 - Precalculus Mathematics for Calculus

Question

A parallelogram has sides of lengths 3 and 5 and one angle is \(50^{\circ} .\) Find the lengths of the diagonals.

Step-by-Step Solution

Answer

Diagonal lengths are approximately 3.836 and 7.714.

Step 1: Understanding the Problem

We need to find the lengths of the diagonals of a parallelogram with sides of lengths 3 and 5, and one angle measuring \(50^{\circ}\). To accomplish this, we will utilize the diagonal formula in a parallelogram that involves sides and an angle between them.

Step 2: Applying the Law of Cosines for Diagonal Calculation

For a parallelogram, the formula for the diagonal \(d_1\) in terms of two adjacent sides \(a, b\) and the included angle \(\theta\) is:\[d_1 = \sqrt{a^2 + b^2 - 2ab \cos\theta}\]Plugging in the values, we get:\[d_1 = \sqrt{3^2 + 5^2 - 2 \times 3 \times 5 \times \cos(50^{\circ})} \approx \sqrt{9 + 25 - 30 \times \cos(50^{\circ})}\]Calculate the cosine and further simplify.

Step 3: Calculating Cosine Value

Calculate \(\cos(50^{\circ})\) using a calculator which gives approximately \(0.6428\). Substitute this value into our formula:\[d_1 \approx \sqrt{9 + 25 - 30 \times 0.6428} \approx \sqrt{9 + 25 - 19.284}\]

Step 4: Simplify the Diagonal Calculation

Simplify the expression under the square root:\[d_1 \approx \sqrt{14.716}\]Compute the square root to find \(d_1\).

Step 5: Find Length of First Diagonal

Calculate the square root:\[d_1 \approx 3.836\]

Step 6: Apply Law of Cosines for Second Diagonal

Calculate using the supplementary angle:For the second diagonal \(d_2\), use the angle \(180^{\circ} - 50^{\circ} = 130^{\circ}\). The law of cosines for \(d_2\) is:\[d_2 = \sqrt{3^2 + 5^2 + 2 \times 3 \times 5 \times \cos(130^{\circ})} \approx \sqrt{9 + 25 + 30 \times \cos(130^{\circ})}\]

Key Concepts

Law of Cosines

The Law of Cosines is a crucial tool in trigonometry, especially when dealing with non-right triangles. Its main use is to find a missing side or angle in a triangle, making it very handy for problems involving arbitrary angles. The formula states that for any triangle with sides of lengths \(a\), \(b\), and \(c\), opposite angles \(A\), \(B\), and \(C\) respectively, the law is mathematically expressed as:\[ c^2 = a^2 + b^2 - 2ab \cos(C) \]This expands the Pythagorean theorem to apply to any triangle, not just right triangles. It helps find one side of a triangle if you know the other two sides and the included angle.

• When used in a parallelogram, like in this exercise, it assists in calculating the length of the diagonals.
• The diagonals can be seen as two triangles, each sharing a side with the parallelogram, making the Law of Cosines an ideal choice.

To solve for diagonal lengths, input the known values for the sides and angle, and solve for the unknowns.

Parallelogram

A parallelogram is a four-sided figure known for having opposite sides that are equal and parallel. This shape forms two sets of parallel lines and each of the opposite angles of a parallelogram are equal too. Features of a Parallelogram:

• The sum of the interior angles is always \(360^{\circ}\).
• It includes specific cases like squares and rectangles, where additional properties hold.

In the context of this exercise, knowing the properties of a parallelogram allows us to understand the way angles and sides interact. For example, the given angle and side lengths can help use the Law of Cosines effectively to find the lengths of diagonals, which split the parallelogram into two congruent triangles. The diagonal effectively acts as a base for these triangles.

Diagonals in Geometry

Diagonals are important line segments in geometry that connect opposite corners in a polygon. In a parallelogram, diagonals play a significant role, as they not only divide the shape into two congruent parts but also offer insight into the figure's properties. Key Aspects:

• Diagonals assist in calculating areas and understanding symmetries.
• They can intersect each other, in the case of a parallelogram, at right angles in specific scenarios like a rhombus or square.

In parallelograms, the diagonals are vital because, using trigonometry and the Law of Cosines, their lengths can be determined with known side lengths and angles. In the problem at hand, each diagonal creates two triangles with the parallelogram's sides. By applying the Law of Cosines, you can solve for the diagonal lengths based on the given angles, illustrating their utility in geometric calculations. Moreover, calculating with diagonals enhances understanding of the parallelogram's entire shape and symmetry.