Question

If two sides of a triangle have the following measures, find the range of possible measures for the third side. 17,8

Short answer

The third side must be greater than 9 and less than 25.

Step-by-step solution

Understand the Triangle Inequality Theorem

The Triangle Inequality Theorem states that the sum of any two sides of a triangle must be greater than the third side. This principle will guide us in determining the possible measures for the third side of the triangle when the other two sides are known.

Apply the Triangle Inequality for Side Combinations

We need to consider each of the side combinations for the two given sides and the third side, which we'll call \( x \): 1. The sum of the two given sides must be greater than \( x \): \[17 + 8 > x\] 2. Adding the first given side and \( x \) should be greater than the second given side: \[17 + x > 8\] 3. Adding the second given side and \( x \) should be greater than the first given side: \[8 + x > 17\]

Solve the Inequality Equations

Now we'll solve each inequality for \( x \):1. \( 17 + 8 > x \) simplifies to \( x < 25 \).2. \( 17 + x > 8 \) simplifies to \( x > -9 \). However, \( x \) must be positive, so this condition is trivially satisfied.3. \( 8 + x > 17 \) simplifies to \( x > 9 \).

Determine the Range for the Third Side

Considering the valid inequalities, we have \( x > 9 \) and \( x < 25 \). Thus, the measure for the third side must satisfy the range:\[ 9 < x < 25 \].

Key concepts

Understanding Key Geometry Concepts

Geometry, the branch of mathematics concerned with the properties and relations of points, lines, surfaces, and solids, offers many interesting challenges. For instance, when analyzing triangles, a fundamental rule to grasp is the Triangle Inequality Theorem. This principle states that any two sides of a triangle must have a sum that is greater than the third side.
The Triangle Inequality Theorem ensures that a set of three lengths forms a triangle. When it comes to solving problems involving triangular measurements, particularly the possible lengths of a triangle's sides, this theorem becomes highly useful. By adhering to this theorem, we can determine the constraints required to maintain the triangle's structural integrity. This is why understanding geometric concepts like this theorem is essential in mathematics.

The Role of Mathematics Education

Mathematics education plays a vital role in helping students comprehend fundamental concepts such as the Triangle Inequality Theorem. Essential mathematical principles are not just about rote learning; they're about fostering a deeper understanding.
Effective mathematics education focuses on:

• Building strong conceptual foundations by explaining core ideas clearly.
• Using meaningful examples, like the Triangle Inequality Theorem, to highlight application.
• Providing practice problems to reinforce concepts.

By integrating these methods, educators can aid students in mastering complex concepts. This empowers students to approach problems analytically, fosters critical thinking, and enhances overall mathematical literacy.

Foundations of Solving Inequalities

Solving inequalities is an important aspect of mathematics that involves finding the range of values that satisfy a particular condition. In the context of geometry, solving inequalities helps determine the potential measurements of a triangle's side.
When you apply the Triangle Inequality Theorem, you essentially set up a series of inequalities based on the possible lengths of a triangle's sides. Each inequality highlights a comparison between the sum of two sides and the remaining side. This problem-solving technique is fundamental when dealing with geometric figures.
Solving these inequalities involves simplifying the given expressions to determine possible values for the third side. For instance, in our exercise, you had to consider the inequalities: 17 + 8 > x, 17 + x > 8, and 8 + x > 17. Simplifying these tells us that the third side, x, must be a length between 9 and 25. By mastering solving inequalities, you gain a strong toolkit for analyzing not only triangles but many other mathematical constructs.