Problem 40 - 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

The diagonals are approximately 3.84 and 7.3.

Step 1: Identify the Given Information

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

Step 2: Apply the Law of Cosines

For a parallelogram with sides \(a\) and \(b\), and a non-inclusive angle \( \theta \), the diagonals can be found using the law of cosines.For diagonal 1 \(d_1\): \[d_1 = \sqrt{a^2 + b^2 - 2ab \cos(\theta)}\]For diagonal 2 \(d_2\): \[d_2 = \sqrt{a^2 + b^2 + 2ab \cos(\theta)}\]

Step 3: Calculate Diagonal 1

Using the sides \( a = 3 \), \( b = 5 \), and angle \( \theta = 50^{\circ} \), we calculate diagonal 1:\[d_1 = \sqrt{3^2 + 5^2 - 2 \times 3 \times 5 \times \cos(50^{\circ})} \]Calculate:\[d_1 = \sqrt{9 + 25 - 30 \cos(50^{\circ})} \approx \sqrt{34 - 19.28} \approx \sqrt{14.72} \approx 3.84 \]

Step 4: Calculate Diagonal 2

Using the same values for \(a\) and \(b\), we proceed to calculate diagonal 2:\[d_2 = \sqrt{3^2 + 5^2 + 2 \times 3 \times 5 \times \cos(50^{\circ})} \]Calculate:\[d_2 = \sqrt{9 + 25 + 30 \cos(50^{\circ})} \approx \sqrt{34 + 19.28} \approx \sqrt{53.28} \approx 7.3 \]

Step 5: Conclusion

The lengths of the diagonals of the parallelogram are approximately 3.84 and 7.3.

Key Concepts

Law of Cosines

The Law of Cosines is an essential tool when dealing with not only triangles but also parallelograms. It helps us determine the lengths of the sides or diagonals when we know angles and other side lengths. This law states that for any triangle with sides lengths of \(a\), \(b\), and \(c\), and an angle \(\theta\) opposite to side \(c\), the equation is:\[c^2 = a^2 + b^2 - 2ab \cos(\theta)\]In a parallelogram, you essentially work with two triangles that share a pair of adjacent sides and an internal angle. When finding and calculating the lengths of diagonals, the formula is slightly adapted according to whether you're looking at Constructive or Deconstructive interference of the angle, as shown with the negative and positive versions in our example.

Diagonals of a Parallelogram

In a parallelogram, the diagonals intersect at their midpoints and split the shape into two congruent triangles. Unlike in a rectangle, the diagonals of a parallelogram are not necessarily equal. To find these diagonal lengths when provided with side lengths and an internal angle, we use the Law of Cosines. For instance, with sides lengths of \(3\) and \(5\) and an angle of \(50^{\circ}\), the diagonals cut through this angle. The equations derived from the Law of Cosines allow us to calculate accurately without needing further geometric constructions. Therefore, for a parallelogram:- Diagonal 1 (\(d_1\)) is calculated with: \[d_1 = \sqrt{a^2 + b^2 - 2ab \cos(\theta)}\]- Diagonal 2 (\(d_2\)) is found with: \[d_2 = \sqrt{a^2 + b^2 + 2ab \cos(\theta)}\]

Angle Calculations

Calculating angles is an essential step in finding the lengths of diagonals in a parallelogram. Knowing one angle (\(50^{\circ}\) in our problem) gives us insight into the entire shape's geometry. Because a parallelogram's internal angles sum up to \(360^{\circ}\), each pair of opposite angles are congruent.Using this known angle, we affectively evaluate the impact on side lengths through trigonometric calculations such as those in the Law of Cosines. For instance, by inputting the given \(50^{\circ}\) in our specific scenario, it allows calculation adjustments in context with the cosine of the angle affecting the respective sides and diagonal. Understanding the angle’s role helps correctly apply formulas and achieve accurate measurements for the diagonals.