Problem 38 - Algebra and Trigonometry

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

Diagonals are approximately 7.297 and 3.836 units.

Step 1: Understanding the Problem

We are given a parallelogram with sides of 3 units and 5 units, and one angle measuring \(50^{\circ}\). We need to find the lengths of the diagonals using these values.

Step 2: Formula for Diagonals

In a parallelogram, the diagonals can be found using the formulas: \( d_1 = \sqrt{a^2 + b^2 + 2ab\cos(\theta)} \) for one diagonal, and \( d_2 = \sqrt{a^2 + b^2 - 2ab\cos(\theta)} \), where \( a \) and \( b \) are the sides, and \( \theta \) is the angle between them.

Step 3: Calculate First Diagonal

Using \( a = 3 \), \( b = 5 \), and \( \theta = 50^{\circ} \), substitute into the first formula: \[ d_1 = \sqrt{3^2 + 5^2 + 2 \times 3 \times 5 \times \cos(50^{\circ})} \] Calculate the cosine and solve for \( d_1 \).

Step 4: Perform Calculation for First Diagonal

First, calculate \(3^2 + 5^2 = 9 + 25 = 34\). Then find \(2 \times 3 \times 5 = 30\) and \(\cos(50^{\circ}) \approx 0.6428\). So the expression becomes:\[ d_1 = \sqrt{34 + 30 \times 0.6428} = \sqrt{34 + 19.284} = \sqrt{53.284} \approx 7.297 \]

Step 5: Calculate Second Diagonal

Now use the second diagonal formula: \[ d_2 = \sqrt{3^2 + 5^2 - 2 \times 3 \times 5 \times \cos(50^{\circ})} \] Substitute the known values and solve for \( d_2 \).

Step 6: Perform Calculation for Second Diagonal

We already have \(3^2 + 5^2 = 34\). The expression becomes: \[ d_2 = \sqrt{34 - 19.284} = \sqrt{14.716} \approx 3.836 \]

Step 7: Summarizing the Results

The lengths of the diagonals are approximately \(7.297\) units and \(3.836\) units, corresponding to \( d_1 \) and \( d_2 \).

Key Concepts

Diagonal Lengths in a Parallelogram

Understanding how to calculate the lengths of diagonals in a parallelogram is essential in geometry. A parallelogram is a quadrilateral with opposite sides that are parallel and equal in length. The diagonals in a parallelogram intersect each other but are not necessarily of equal length. By knowing the length of the sides and one angle, we can determine the lengths of the diagonals. This involves the use of trigonometric concepts and specific formulas. The formula for diagonal length in a parallelogram combines the side lengths and an interior angle, offering a systematic method to obtain this measurement.

Applying the Cosine Law

The cosine law is a fundamental concept in trigonometry, essential for solving many problems related to triangles and parallelograms. It works well with non-right triangles, providing a connection between a triangle's sides and its angles. For a parallelogram, the cosine law is modified to find the diagonal lengths as the sides form two triangles with the diagonals.

• First diagonal formula: \[ d_1 = \sqrt{a^2 + b^2 + 2ab\cos(\theta)} \]
• Second diagonal formula: \[ d_2 = \sqrt{a^2 + b^2 - 2ab\cos(\theta)} \]

By substituting the known values into these formulas, we can determine the diagonal lengths. This method leverages the measurement of the angle to adjust the equation, showcasing the versatility of the cosine law in various geometrical configurations.

Understanding Angle Measurement

Angle measurement is a critical component when determining diagonal lengths in a parallelogram. The interior angle is an angle inside the parallelogram, affecting the calculation of the diagonals because it informs how the sides interact with the diagonals. In our example, a given angle of \(50^{\circ}\) helps find the appropriate length of each diagonal. When dealing with angles, especially in trigonometric formulas, it's crucial to recognize that angles must be provided in degrees unless specified otherwise. It's also important to use a calculator that can compute cosine values accurately to ensure precision when inputting these angles into your formula calculations.

Key Geometry Formulas for Parallelograms

Geometry involves various formulas that express relationships between shapes' sides, angles, and other elements. When focusing on parallelograms, some key formulas include those for area, perimeter, and most importantly, diagonal lengths.

• Diagonal formulas: \[ d_1 = \sqrt{a^2 + b^2 + 2ab\cos(\theta)} \]\[ d_2 = \sqrt{a^2 + b^2 - 2ab\cos(\theta)} \]
• Area formula: \[ \text{Area} = a \times b \times \sin(\theta)\]

Understanding these formulas allows for comprehensive calculations of a parallelogram's properties. They play a crucial role in determining fundamental characteristics such as area and diagonal measurements, leveraging the principles of trigonometry and geometry to solve practical problems.