Problem 54 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. The sides of a parallelogram are 4.0 centimeters and 6.0 centimeters. One angle is \(58^{\circ}\) and another is \(122^{\circ} .\) Find the lengths of the diagonals of the parallelogram.

Step-by-Step Solution

Answer

The diagonals are approximately 5.155 cm and 8.798 cm.

Step 1: Identify the Known Elements

We know the sides of the parallelogram are 4.0 cm and 6.0 cm, and the angles are \(58^{\circ}\) and \(122^{\circ}\). The angle between the two diagonals is the same as the given angle, which is \(58^{\circ}\).

Step 2: Use the Diagonal Length Formula for Parallelograms

The length of a diagonal in a parallelogram can be found using the formula \(d = \sqrt{a^2 + b^2 - 2ab\cos(C)}\). This formula is derived from the law of cosines, where \(a\) and \(b\) are the lengths of the sides and \(C\) is the angle between them.

Step 3: Calculate the First Diagonal

For the first diagonal, use sides 4.0 cm and 6.0 cm with the angle \(58^{\circ}\). Substitute into the formula:\[d_1 = \sqrt{4.0^2 + 6.0^2 - 2 \times 4.0 \times 6.0 \times \cos(58^{\circ})}\]Calculate:\[d_1 = \sqrt{16 + 36 - 48 \cos(58^{\circ})}\]\[d_1 = \sqrt{52 - 48 \times 0.5299}\]\[d_1 = \sqrt{52 - 25.4352}\]\[d_1 = \sqrt{26.5648}\]\[d_1 \approx 5.155 \, \text{cm}\]

Step 4: Calculate the Second Diagonal

For the second diagonal, use the same sides 4.0 cm and 6.0 cm but the angle \(122^{\circ}\). Substitute into the formula:\[d_2 = \sqrt{4.0^2 + 6.0^2 - 2 \times 4.0 \times 6.0 \times \cos(122^{\circ})}\]Calculate:\[d_2 = \sqrt{16 + 36 - 48 \cos(122^{\circ})}\]\[d_2 = \sqrt{52 - 48 \times (-0.5299)}\]\[d_2 = \sqrt{52 + 25.4352}\]\[d_2 = \sqrt{77.4352}\]\[d_2 \approx 8.798 \, \text{cm}\]

Key Concepts

Diagonal Length Calculation

Parallelograms have special properties when it comes to their diagonals. Calculating the length of a diagonal in a parallelogram is a vital skill in geometry. To find diagonal lengths, we use a specific formula: \[ d = \sqrt{a^2 + b^2 - 2ab\cos(C)} \] Here, each diagonal is treated as if it is creating a triangle with the two sides of the parallelogram and the included angle (either \(58^{\circ}\) or \(122^{\circ}\)). This method is handy because it allows us to use our knowledge of trigonometry to solve geometric problems easily.

• Step 1: Identify the sides and the relevant angle for the diagonal in question. In this task, the sides are 4 cm and 6 cm.
• Step 2: Use the diagonal formula to calculate each diagonal length.

Understanding how to calculate diagonal lengths in parallelograms not only enriches your geometry skills but also sharpens your problem-solving abilities in various mathematical contexts.

Law of Cosines

The Law of Cosines is an essential concept in trigonometry and geometry. It helps in calculating unknown lengths or angles in triangles, especially when dealing with non-right triangles, like the ones formed by a parallelogram's diagonals. This law can be stated as: \[ c^2 = a^2 + b^2 - 2ab \cos(C) \] where \( c \) is the length of the side opposite the angle \( C \), and \( a \) and \( b \) are the lengths of the other two sides.

• This formula is the key to finding the lengths of the diagonals in a parallelogram when you know the side lengths and the angles.
• It acts like an extension of the Pythagorean theorem, useful for any type of triangle, not just right ones.

By applying the Law of Cosines in the context of parallelograms, we can understand and solve complex geometric shapes with ease.

Angle Measurement

In geometry, understanding and accurately measuring angles is crucial, especially in parallelograms where angles determine many of their properties. In this exercise, you worked with angles \(58^{\circ}\) and \(122^{\circ}\). Each pair of opposite angles in a parallelogram are equal, and adjacent angles are supplementary. This means they add up to \(180^{\circ}\).

• For example, if one angle is \(58^{\circ}\), the angle opposite to it must also be \(58^{\circ}\), while its adjacent angles will be \(122^{\circ}\) (since \(58^{\circ} + 122^{\circ} = 180^{\circ}\)).
• When calculating diagonal lengths, knowing the exact angle helps apply the correct trigonometric calculations accurately.

Fundamentally, understanding angles aids greatly in solving geometrical problems, including those involving diagonals of parallelograms.

Trigonometric Calculations

Trigonometry provides powerful tools for solving geometric problems involving angles and sides, such as those found in this parallelogram exercise. At the core, the calculations rely on the properties of sine, cosine, and tangent functions.

• Cosine Function: For both diagonals, the cosine of the given angles (\(58^{\circ}\) and \(122^{\circ}\)) was used. It allows us to find the length of a side when an angle and two sides are known.
• Positive and Negative Values: Cosine values change based on the angle. For angles greater than \(90^{\circ}\), like \(122^{\circ}\), the cosine becomes negative, which impacts the diagonal length calculations.

These trigonometric calculations are essential when dealing with non-right angles, transforming our approach to geometric challenges and enhancing precision in measurements.