Problem 65 - Precalculus with Limits

Question

WRITING A triangle has side lengths of 10 centimeters, 16 centimeters, and 5 centimeters. Can the Law of Cosines be used to solve the triangle? Explain.

Step-by-Step Solution

Answer

No, the Law of Cosines cannot be used to solve the triangle with only the given side lengths, because there's no included angle known.

Step 1 - Understand the Law of Cosines

The Law of Cosines states that: \(c^2 = a^2 + b^2 - 2ab cos(C)\), where a, b, c are the sides of a triangle, and C is the included angle. For a triangle with side lengths 10 cm, 16 cm, and 5 cm, assign the longest side (c) as 16, and the others (a,b) can be either 10 or 5.

Step 2 - Identify the Knowns and Unknowns

In order to apply the law of cosines, one of the angles and its opposite side must be known. However, in the given problem, only the sides of the triangle are given but no angle is provided. Therefore, the law of cosines cannot be directly applied because the cosine of the included angle between any two sides is unknown.

Step 3 - Determine if the Law of Cosines can be used

The Law of Cosines can be used to solve a triangle when either two sides and the included angle are known, or when all three sides are known, and it's required to find an angle. In the given situation, neither is the case. Therefore, the answer is no, the Law of Cosines cannot be used to solve the triangle with only the three side lengths given, as we do not have enough information.

Key Concepts

Triangle Side Lengths

Understanding the side lengths of a triangle is crucial when diving into geometry, especially when using the Law of Cosines. The side lengths dictate not only the shape but what can be calculated about the triangle.
To gauge potential calculations like angles or other sides, side lengths of a triangle should satisfy the triangle inequality theorem. This theorem posits that for any triangle, the sum of the lengths of any two sides should be greater than the length of the remaining side. For instance, the triangle with sides 10 cm, 16 cm, and 5 cm adheres to this rule because:

• 10 + 5 = 15, which is less than 16
• 10 + 16 = 26, which is greater than 5
• 16 + 5 = 21, which is greater than 10

Using these side lengths, one can embark on solving the triangle for additional properties if more information, such as angles, becomes available.

Geometry Problem Solving

When encountering a geometry problem, the approach and steps taken can significantly influence the outcome. Strategizing is vital whether figuring out angles, side lengths, or areas.
It's essential to:

• Begin by listing known values, such as side lengths or angles.
• Determine what you need to find, like another side, an angle, or the area.
• Decide which mathematical tools or theorems best fit the problem. For example, the Pythagorean Theorem for right triangles or the Law of Cosines for non-right triangles.

For this exercise, though the Law of Cosines was considered, it wasn't directly applicable due to missing angles. This highlights a fundamental problem-solving strategy: choose the right tool or strategy for the problem at hand. In situations with only side lengths, consider other geometric principles or supplementary data like angles.

Angle Calculation

Calculating angles in a triangle involves understanding and applying various mathematical theorems and laws. The Law of Cosines is particularly useful in non-right triangles where some angles are unknown.
The formula is given by:\[ c^2 = a^2 + b^2 - 2ab \cos(C) \]This allows for calculating angle C when sides a, b, and c are known. It's applicable if you're trying to solve for a missing angle given each side length.
In the current exercise, although we have the sides, we can't directly calculate the angles without a known or measurable angle to begin with. In absence of such data, other methods or additional information would be required, reinforcing the importance of comprehending how each part of the triangle relates to each other.
Thus, when faced with only side lengths, look to potential external means to determine an angle, then use the Law of Cosines efficiently.