Problem 47 - Algebra 2

Question

Geometry The lengths of the adjacent sides of a parallelogram are 54 \(\mathrm{cm}\) and 78 \(\mathrm{cm}\) . The larger angle measures \(110^{\circ} .\) What is the length of the longer diagonal? Round your answer to the nearest centimeter.

Step-by-Step Solution

Answer

The length of the longer diagonal of the parallelogram, when rounded to the nearest centimeter, is approximately 122 cm.

Step 1: Identify Given Values

The given values from the exercise are: length of one side (\(a\)) = 54cm, length of the other side (\(b\)) = 78cm and one angle \(\theta = 110^{\circ}\). These will be the variables for the formula.

Step 2: Convert Degrees to Radians

Angles should be converted to radians before using them in trigonometric functions in most calculators. Therefore, convert \(110^{\circ}\) into radians. Use the relationship that \(180^{\circ}\) equals \(\pi\) radians. So, \(\theta = 110 * (\pi / 180) \approx 1.92\) rad.

Step 3: Substitute In The Formula

Substitute the given values into the formula for the diagonal \(d = \sqrt{a^2 + b^2 + 2ab\cos\theta}\). Then, \(d = \sqrt{(54)^2 + (78)^2 + 2*54*78*\cos(1.92)}.\)

Step 4: Calculate The Result

Calculate the square root and round the result to the nearest centimeter as dictated by the problem. Given the rounding requirement, the nearest whole number should be used.

Key Concepts

Parallelogram

A parallelogram is a four-sided shape where opposite sides are parallel and equal in length. This means:

• Opposite sides are equal: If one side is 'a' and the other 'b', the opposite sides will all measure 'a' or 'b'.
• Opposite angles are equal: Angles facing each other are the same.
• Adjacent angles: These add up to 180 degrees because the sides are parallel, creating supplementary angles.

Parallelograms are important in geometry due to their symmetrical properties. They are used to explore diagonal lengths, areas, and more. Understanding a parallelogram's properties helps when applying trigonometric functions, especially when calculating diagonal lengths.

Trigonometry

Trigonometry deals with the relationships between the angles and sides of triangles, and by extension, other polygonal shapes like parallelograms. Key concepts include:

• Sine, Cosine, and Tangent: These functions relate angles to side lengths.
• Cosine Rule: Often used for finding unknown side lengths, especially in non-right triangles. The formula is: \[ c = \sqrt{a^2 + b^2 - 2ab\cos(\theta)} \]
• Application in Parallelograms: Here, the diagonals can be calculated using variations of these rules.

In the problem, the cosine of an angle was crucial for finding the longer diagonal's length. This illustrates the practical application of trigonometry in geometrical shapes.

Diagonals

Diagonals in a parallelogram are significant as they connect opposite vertices and can provide insights into the shape's symmetry and dimensions. Key points about diagonals:

• Formula for Diagonals: Use trigonometric rules to find lengths, such as using cosine for calculating diagonal distance.
• Intersecting Properties: In a parallelogram, diagonals bisect each other, dividing them into equal halves.

Finding diagonal lengths helps in understanding the parallelogram's geometry. For instance, in the exercise, the diagonal length required manipulating side lengths and angles through trigonometric operations.

Radians

Radians are a unit for measuring angles, essential for using angles in trigonometric calculations. Unlike degrees, radians provide a direct relationship between the length of an arc and the radius of the circle. Important aspects include:

• Conversion: 180 degrees is equivalent to \( \pi \) radians. Converting from degrees to radians involves multiplying by \( \pi/180 \).
• Why Use Radians: Many mathematical formulas, especially those involving calculus or trigonometry, work more naturally with radians.

In the exercise, converting 110 degrees to radians was necessary for accurate trigonometric computation. This conversion ensures that calculators and formulas yield correct results when involving trigonometric functions like cosine.