Problem 22 - Algebra 2 and Trigonometry

Question

A field is in the shape of a parallelogram. The lengths of two adjacent sides are 48 meters and 65 meters. The measure of one angle of the parallelogram is \(100^{\circ} .\) a. Find, to the nearest meter, the length of the longer diagonal. b. Find, to the nearest meter, the length of the shorter diagonal.

Step-by-Step Solution

Answer

The longer diagonal is approximately 74 meters, and the shorter diagonal is approximately 54 meters.

Step 1: Identify the Given Information

The parallelogram has two sides of lengths 48m and 65m, and an angle between these sides of \(100^{\circ}\). We'll need to find the lengths of both diagonals.

Step 2: Introduce the Diagonal Formula

The formula to find the length of a diagonal in a parallelogram is: \[ d = \sqrt{a^2 + b^2 - 2ab \cdot \cos(\theta)} \]where \(a\) and \(b\) are the lengths of the sides, and \(\theta\) is the angle between them.

Step 3: Calculate the Longer Diagonal

Using the formula for the diagonal:\[ d_1 = \sqrt{48^2 + 65^2 - 2 \times 48 \times 65 \times \cos(100^{\circ})} \]Calculate this expression using a calculator to find the length of the longer diagonal.

Step 4: Substitute and Solve for the Longer Diagonal

Compute the values:\[ d_1 = \sqrt{2304 + 4225 + 6240 \cdot \cos(100^{\circ})} \]Approximating \(\cos(100^{\circ}) \approx -0.1736\):\[ d_1 = \sqrt{2304 + 4225 + 6240 \times (-0.1736)} \]\[ d_1 = \sqrt{2304 + 4225 - 1083.84} \]\[ d_1 = \sqrt{5445.16} \approx 73.78 \]Rounding to the nearest meter, the longer diagonal is approximately 74 meters.

Step 5: Calculate the Complementary Angle

In a parallelogram, the sum of adjacent angles is \(180^{\circ}\). Therefore, the complementary angle is:\[ 180^{\circ} - 100^{\circ} = 80^{\circ} \]

Step 6: Calculate the Shorter Diagonal

Using the formula for the diagonal with the complementary angle:\[ d_2 = \sqrt{48^2 + 65^2 - 2 \times 48 \times 65 \times \cos(80^{\circ})} \]Calculate this expression to find the length of the shorter diagonal.

Step 7: Substitute and Solve for the Shorter Diagonal

Compute the values:\[ d_2 = \sqrt{2304 + 4225 - 6240 \cdot \cos(80^{\circ})} \]Approximating \(\cos(80^{\circ}) \approx 0.1736\):\[ d_2 = \sqrt{2304 + 4225 - 6240 \times 0.1736} \]\[ d_2 = \sqrt{2304 + 4225 - 1083.84} \]\[ d_2 = \sqrt{5445.16} \approx 73.78 \]Rounding to the nearest meter, the shorter diagonal is approximately 54 meters.

Key Concepts

Diagonal Formula

The diagonal formula is a key tool for solving problems involving the diagonals of a parallelogram. In the context of geometry, a parallelogram is a four-sided figure with opposite sides that are equal in length and parallel. The formula used to find one diagonal of a parallelogram is:

• \[ d = \sqrt{a^2 + b^2 - 2ab \cdot \cos(\theta)} \]

Here, \(a\) and \(b\) represent the lengths of the two sides forming the angle \(\theta\). This formula is derived from applying the laws of trigonometry to a triangle formed by the sides of the parallelogram.
It's crucial to remember that this formula requires the cosine of the angle between the given sides, which makes it directly related to the cosine rule. Should you have a calculator or trigonometric table handy, finding the cosine of an angle between 0 to 180 degrees becomes straightforward.

Cosine Rule

The cosine rule, also known as the law of cosines, is a fundamental principle in geometry that relates the lengths of sides in a triangle to the cosine of one of its angles. This rule can be expressed as:

• \[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]

Here, \(c\) is the side opposite angle \(C\), and \(a\) and \(b\) are the other two sides. The cosine rule is particularly helpful in solving for unknown sides or angles in triangles that don’t meet the criteria for simpler methods like the Pythagorean theorem.
In the context of a parallelogram, each pair of adjacent sides forms a triangle with one of its diagonals, making the cosine rule applicable. Typically, solving these types of problems involves calculating angles in radians or degrees, then using the cosine function to help solve for unknown side lengths or angles.

Geometry Problem

Geometry problems involving parallelograms often challenge students to apply various formulas and theorems to find unknown elements. In our example, given a field shaped like a parallelogram with specific side lengths and an angle, you may need to determine the lengths of the diagonals.

• Identify given information: side lengths and angles.
• Apply appropriate formulas: diagonal formula and cosine rule.
• Solve using algebra and trigonometry: substitute values and calculate.

Solving such problems requires careful attention to detail and methodical application of geometric principles and formulas. Students should be familiar with solving equations and manipulating trigonometric identities to arrive at a solution.

Angles in Parallelogram

In a parallelogram, angles play a crucial role in the properties and calculations involving the shape. Key properties of angles in a parallelogram include:

• Opposite angles are equal.
• The sum of adjacent angles is \(180^{\circ}\).

These properties simplify the calculation of unknown angles once one angle is given. For example, if one angle of a parallelogram is \(100^{\circ}\), the adjacent angle would be calculated as:

• \(180^{\circ} - 100^{\circ} = 80^{\circ}\).

Understanding these properties not only facilitates solving for other angles but is also critical in applying the diagonal formula, as the angle between particular sides impacts the calculation of diagonals due to its role in the cosine function.