Finding The Diagonal Of A Parallelogram Using The Law Of Cosines

In this article, we will explore how to find the length of a diagonal in a parallelogram when given the side lengths and one angle. We'll use the Law of Cosines, a fundamental trigonometric principle, to solve this problem. This is a common problem in geometry and trigonometry, and understanding the steps involved is crucial for anyone studying these fields. Let's dive into the specifics of the problem and how we can solve it.

Understanding the Problem

Let's first restate the problem clearly. A parallelogram has side lengths of 13 and 17 units, and one of its angles measures 64 degrees. Our goal is to find the length of one of the diagonals, which we'll call x, rounded to the nearest whole number. This problem combines geometric properties of parallelograms with trigonometric principles, specifically the Law of Cosines.

To effectively solve this, we need to visualize the parallelogram and understand its properties. A parallelogram has opposite sides that are equal in length and parallel, and opposite angles that are equal. However, adjacent angles are supplementary, meaning they add up to 180 degrees. The diagonals of a parallelogram bisect each other, but they are not necessarily equal in length unless the parallelogram is a rectangle.

Given the side lengths and an angle, we can use the Law of Cosines to find the length of the diagonal that forms a triangle with these sides and the given angle. It's essential to recognize that a parallelogram has two diagonals, and depending on which angle we use, we will find a different diagonal length. In this case, since we're given a 64-degree angle, we will find the diagonal opposite this angle. Understanding these geometric nuances is crucial before we apply the Law of Cosines.

The Law of Cosines: A Key Tool

The Law of Cosines is a generalization of the Pythagorean theorem and is a powerful tool for solving triangles when we don't have a right angle. It relates the lengths of the sides of a triangle to the cosine of one of its angles. The formula is given by:

$a^2 = b^2 + c^2 - 2bc \* cos(A)$

where:

• a is the length of the side opposite angle A.
• b and c are the lengths of the other two sides.
• A is the angle opposite side a.

In our parallelogram problem, we can consider the diagonal as one side of a triangle formed by two sides of the parallelogram and the given angle. By applying the Law of Cosines, we can set up an equation to solve for the length of the diagonal. Identifying the correct correspondence between the sides and angles is crucial for accurate calculations.

It's important to remember that the Law of Cosines is particularly useful when we know two sides and the included angle (the angle between those sides) or when we know all three sides of a triangle. In this problem, we are given two sides of the parallelogram (13 and 17) and the included angle (64 degrees), making the Law of Cosines the perfect tool for finding the diagonal. Now, let's apply this law to our specific problem.

Applying the Law of Cosines to the Parallelogram

Now, let's apply the Law of Cosines to our parallelogram problem. We have side lengths of 13 and 17, and an angle of 64 degrees. We want to find the length of the diagonal, which we'll call x. We can set up the equation as follows:

$x^2 = 13^2 + 17^2 - 2 \* 13 \* 17 \* cos(64^{\circ})$

This equation directly applies the Law of Cosines, where x is the side opposite the 64-degree angle, and 13 and 17 are the other two sides. The next step is to calculate the values and solve for x. We'll need a calculator to find the cosine of 64 degrees and then perform the arithmetic operations.

It is essential to double-check that your calculator is in degree mode when calculating the cosine of 64 degrees. Using radian mode instead of degree mode will result in a completely different answer. After ensuring the correct mode, calculate cos(64°) and then proceed with the multiplication and subtraction in the equation. Remember to take the square root of the final result to find the value of x. This methodical approach will help ensure accuracy in the calculations.

Solving for x: Step-by-Step Calculation

Let's break down the calculation step by step to find the value of x. First, we calculate the squares of the side lengths:

$13^2 = 169$ $17^2 = 289$

Next, we find the cosine of 64 degrees. Using a calculator, we get:

$cos(64^{\circ}) ≈ 0.438$

Now, we substitute these values into the Law of Cosines equation:

$x^2 = 169 + 289 - 2 \* 13 \* 17 \* 0.438$

Perform the multiplication:

$2 \* 13 \* 17 \* 0.438 ≈ 194.4$

Substitute this back into the equation:

$x^2 = 169 + 289 - 194.4$

Now, add and subtract:

$x^2 = 458 - 194.4$ $x^2 = 263.6$

Finally, take the square root of both sides to find x:

$x = \sqrt{263.6} ≈ 16.24$

Since we need to round the answer to the nearest whole number, x is approximately 16. This step-by-step calculation demonstrates the application of the Law of Cosines and the arithmetic operations required to arrive at the solution. Always remember to double-check each step to ensure accuracy.

The Other Diagonal

Now that we've found one diagonal, it's insightful to consider the other diagonal of the parallelogram. Recall that parallelograms have two diagonals, and their lengths are generally different unless the parallelogram is a rectangle or a square. We used the 64-degree angle to find the length of one diagonal. To find the length of the other diagonal, we need to consider the supplementary angle to 64 degrees.

Since adjacent angles in a parallelogram are supplementary, the angle adjacent to the 64-degree angle is 180° - 64° = 116°. We can use this angle and the same side lengths (13 and 17) in the Law of Cosines to find the length of the other diagonal. This exercise highlights the importance of understanding the properties of parallelograms and how they influence the application of trigonometric principles. Let's calculate the length of this other diagonal for comparison and a deeper understanding of the problem.

Calculating the Other Diagonal's Length

To find the length of the other diagonal, let's call it y, we'll use the supplementary angle of 64 degrees, which is 116 degrees. We apply the Law of Cosines again:

$y^2 = 13^2 + 17^2 - 2 \* 13 \* 17 \* cos(116^{\circ})$

We already know that:

$13^2 = 169$ $17^2 = 289$

Now, we need to find the cosine of 116 degrees. Using a calculator:

$cos(116^{\circ}) ≈ -0.438$

Notice that the cosine of 116 degrees is negative. This is because 116 degrees is an obtuse angle (greater than 90 degrees), and the cosine function is negative in the second quadrant. Now, substitute these values into the Law of Cosines equation:

$y^2 = 169 + 289 - 2 \* 13 \* 17 \* (-0.438)$

Perform the multiplication. Note that multiplying by a negative number will change the sign of the term:

$2 \* 13 \* 17 \* (-0.438) ≈ -194.4$

Substitute this back into the equation:

$y^2 = 169 + 289 - (-194.4)$ $y^2 = 169 + 289 + 194.4$

Now, add the values:

$y^2 = 652.4$

Finally, take the square root of both sides to find y:

$y = \sqrt{652.4} ≈ 25.54$

Rounding to the nearest whole number, y is approximately 26. This calculation shows that the other diagonal is significantly longer than the one we calculated earlier. Comparing the lengths of the two diagonals provides a better understanding of the parallelogram's geometry and the impact of the angles on the diagonal lengths.

Conclusion

In conclusion, we successfully found the length of the diagonal of a parallelogram using the Law of Cosines. We started by understanding the problem, identifying the given information (side lengths and an angle), and recognizing the applicability of the Law of Cosines. We then applied the formula, carefully calculated the values, and solved for the diagonal length. We found one diagonal to be approximately 16 units long. Furthermore, we extended our analysis to calculate the length of the other diagonal using the supplementary angle, finding it to be approximately 26 units long. This comprehensive approach not only answers the initial question but also deepens our understanding of the geometric properties of parallelograms and the use of trigonometric principles to solve real problems.

This exercise demonstrates the power of the Law of Cosines in solving geometric problems, particularly those involving triangles that are not right-angled. By carefully applying the formula and understanding the relationships between sides and angles, we can accurately determine unknown lengths and angles in various geometric figures. Remember to always double-check your calculations and ensure that your calculator is in the correct mode (degrees or radians) to achieve accurate results.