Question

One side of a triangle is 15 cm long, and another side is 28 cm long. Which of the following is a possible length, in centimeters, for the third side? A. 2 B. 12 C. 31 D. 44 E. 52

Short answer

Answer: C) 31 cm

Step-by-step solution

Identify the given information

We are given that one side of the triangle is 15 cm long, and another side is 28 cm long.

Apply the Triangle Inequality Theorem to each of the given options

According to the Triangle Inequality Theorem, for any option to be a valid third side, both of the following conditions must hold: 1. 15 + option > 28 2. 28 + option > 15

Check each of the given options

Option A: 2 1. 15 + 2 = 17, which is not greater than 28 This option doesn't satisfy the Triangle Inequality Theorem, so it cannot be a valid third side. Option B: 12 1. 15 + 12 = 27, which is not greater than 28 This option also doesn't satisfy the Triangle Inequality Theorem, so it cannot be a valid third side. Option C: 31 1. 15 + 31 = 46, which is greater than 28 2. 28 + 31 = 59, which is greater than 15 This option satisfies the Triangle Inequality Theorem, so it can be a valid third side. Option D: 44 1. 15 + 44 = 59, which is greater than 28 2. 28 + 44 = 72, which is greater than 15 However, the sum of the other two sides (15 + 28 = 43) is smaller than 44, so it cannot form a triangle. Therefore, it cannot be a valid third side. Option E: 52 1. 15 + 52 = 67, which is greater than 28 2. 28 + 52 = 80, which is greater than 15 Similar to Option D, the sum of the other two sides (15 + 28 = 43) is smaller than 52, so it cannot form a triangle. Therefore, it cannot be a valid third side.

Conclusion

According to the Triangle Inequality Theorem, the only possible length for the third side of the triangle is 31 cm (Option C).

Key concepts

Understanding Triangle Side Lengths

When we explore the lengths of the sides of a triangle, we enter the fascinating world of Triangle Inequality Theorem. This fundamental principle dictates a set of rules determining whether three lengths can form a triangle.

Let's consider an exercise where side lengths of 15 cm and 28 cm are given for a triangle, and we need to find a possible third side. The lengths of A. 2, B. 12, C. 31, D. 44, and E. 52 cm are the options. Understanding that the combined length of any two sides of a triangle must be greater than the length of the third side, we can systematically evaluate each option.

Key Takeaways for Triangle Side Lengths

• To constitute a triangle, the sums of every pair of sides must be greater than the length of the remaining side.
• Applying the Triangle Inequality Theorem helps in avoiding impossible triangular shapes.
• The relationship among the sides is essential for creating a valid triangle.

User-friendly strategies, such as process of elimination, can simplify these checks significantly for students.

Navigating Geometry Problems

Geometry problems can often seem intimidating, but having a strategy can make them far more approachable. Let's illustrate this by breaking down our triangle problem into manageable steps.

Firstly, identify the known variables. In this case, we have two known side lengths of a triangle. Next, we're applying the Triangle Inequality Theorem to find which of the given lengths can form the third side—a perfect demonstration of geometry in practice.

Effective Strategies for Solving Geometry Problems

• Begin by identifying and listing the known elements of the problem.
• Apply relevant theorems or principles, such as the Triangle Inequality Theorem in this scenario.
• Work systematically through the options with logical reasoning.
• Conclude with the solution that satisfies all the conditions posed by the theorem.

With this structured approach, geometry problems become less daunting and more of a puzzle to unlock.

Applying Mathematical Principles

Mathematical principles are the backbone of problem-solving in mathematics, offering a reliable framework to approach and solve exercises, like determining the third side of a triangle. In our example, the Triangle Inequality Theorem is the principle in action, guiding us to the correct answer.

Impact of Mathematical Principles

• They provide a systematic approach to reach consistent results.
• Mathematical principles like the Triangle Inequality Theorem ensure accuracy in geometry.
• Recognizing and applying these principles is crucial for solving complex problems.

Adhering to mathematical principles not only aids in finding correct solutions but also helps in understanding the logic behind the mathematical concepts, enriching the learning experience and enabling students to tackle more challenging problems.