Problem 48 - Algebra 2

Question

Geometry The lengths of the adjacent sides of a parallelogram are 21 \(\mathrm{cm}\) and 14 \(\mathrm{cm} .\) The smaller angle measures \(58^{\circ} .\) What is the length of the shorter diagonal? Round your answer to the nearest centimeter.

Step-by-Step Solution

Answer

The length of the shorter diagonal is around 16 cm

Step 1: Identify the relevant formula

The formula for the length of the diagonal of a parallelogram when the sides and one angle are given is: \(d = \sqrt{a^2 + b^2 - 2ab*cos(θ)}\), where a and b are the lengths of the adjacent sides and θ is the angle between them.

Step 2: Substitute the given values

Next, plug in the given values into the formula: \(d = \sqrt{21^2 + 14^2 - 2*21*14*cos(58^\circ)}\).

Step 3: Compute the value

Finally, compute the value of d to find the length of the shorter diagonal.

Key Concepts

Understanding the Parallelogram in Geometry

A parallelogram is a four-sided shape where opposite sides are parallel and equal in length. It's one of the fundamental shapes studied in geometry. The properties of a parallelogram play a crucial role in solving various geometric problems. Understanding these properties helps us calculate angles, side lengths, and even the lengths of diagonals.

Key characteristics of a parallelogram include:

• Opposite sides are equal and parallel.
• Adjacent angles add up to 180 degrees.
• Each diagonal divides the parallelogram into two congruent triangles.
• The area can be calculated using base and height or the lengths of sides and sine of the included angle.

With these properties in mind, challenges involving parallelograms become more manageable. They form the foundation for more complex geometric exercises.

Determining Diagonal Length Using Given Formulas

The diagonal of a parallelogram can be determined using a specific formula when you know the side lengths and the included angle. This formula is essential because diagonals of a parallelogram do not necessarily equal each other. They can vary greatly based on the angles and lengths of the sides of the parallelogram.

The formula used in this problem is derived from the law of cosines:\[d = \sqrt{a^2 + b^2 - 2ab \cos(\theta)}\]where:

• \( a \) and \( b \) are the lengths of the adjacent sides of the parallelogram.
• \( \theta \) is the angle between these two sides.

This formula allows us to find the precise length of the diagonals by substituting the known values and performing the calculations. It's a straightforward application of trigonometry and algebra and simplifies what could otherwise be a geometrically complex problem.

Applying Trigonometry in Geometry Problems

Trigonometry frequently appears in geometry, especially when dealing with angles and triangle relationships within geometric shapes like parallelograms. In this particular problem, the use of trigonometry is evident in the application of the cosine function.

The cosine function (\(\cos\)) helps relate the angle of a triangle to the lengths of its sides. In a parallelogram, each diagonal can be thought of as forming two triangles. Knowing one of the angles and the side lengths, we can effectively apply trigonometric principles to calculate unknowns – in this case, the diagonal length.

Here is a brief look at why trigonometry is essential:

• It enables calculations involving angles and sides without directly measuring them.
• Provides solutions to problems involving non-right triangles, like those formed by the diagonals of a parallelogram.
• Facilitates the use of formulas such as the law of cosines, enhancing our problem-solving toolkit.

Thus, understanding how trigonometry fits into geometry is crucial for tackling a wide range of exercises, giving you the tools to solve problems with confidence.