Problem 182 - 1,001 GED Practice Questions For Dummies

Question

If two sides of a triangle are 10.2 and 5.8 inches long, which of the following cannot be the length of the other side? (A) 4.3 inches (B) 5.8 inches (C) 11.7 inches (D) 15.2 inches

Step-by-Step Solution

Answer

The length that cannot be the third side of the triangle with given sides 10.2 and 5.8 inches is: \(4.3\) inches (Option A).

Step 1. Apply the Triangle Inequality Theorem with side lengths

For each possible length of the third side, we need to check that the sum of the lengths of any two sides is greater than the length of the third side.

Step 2. Test each possibility

Let's test each of the given possibilities: a. x = 4.3 inches 10.2 + 5.8 > 4.3 (True) 10.2 + 4.3 > 5.8 (True) 5.8 + 4.3 > 10.2 (False) - This option violates the Triangle Inequality Theorem. b. x = 5.8 inches 10.2 + 5.8 > 5.8 (True) 10.2 + 5.8 > 10.2 (True) 5.8 + 5.8 > 10.2 (True) - No violations. c. x = 11.7 inches 10.2 + 5.8 > 11.7 (True) 10.2 + 11.7 > 5.8 (True) 5.8 + 11.7 > 10.2 (True) - No violations. d. x = 15.2 inches 10.2 + 5.8 > 15.2 (False) 10.2 + 15.2 > 5.8 (True) 5.8 + 15.2 > 10.2 (True) - This option violates the Triangle Inequality Theorem. Now we have tested all the possibilities and found two options that violate the Triangle Inequality Theorem: (a) 4.3 inches and (d) 15.2 inches. However, since there is only one option in the question, the correct answer must be the one that cannot be the length of the third side.

Step 3. Choose the correct option

Based on our tests, the side length that cannot be the third side of the triangle with given sides 10.2 and 5.8 inches is: (a) 4.3 inches

Key Concepts

Understanding Triangle Sides

When working with triangles in geometry, one of the most important things to understand is the relationship between its sides. Each side plays a crucial role in forming a triangle. The sides of a triangle must adhere to certain mathematical principles to form a valid shape. One such principle is the Triangle Inequality Theorem, which ensures geometric validity by posing constraints on side lengths.

To determine if three given lengths can form a triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. This theorem ensures that:

• The sides cannot lie in a straight line, which would prevent the formation of a closed shape.
• If any one triangle inequality fails, the set of lengths cannot form a triangle.

Understanding side lengths not only aids in identifying the possibility of a triangle's existence but also helps solve geometry problems with ease.

Exploring Mathematical Reasoning

Mathematical reasoning involves using logic to solve problems and verify results. It is especially important when applying the Triangle Inequality Theorem. By systematically testing inequalities with potential side lengths, you can determine if a valid triangle can form.

Here's how mathematical reasoning is applied:

• Check all combinations of two sides adding up to more than the third side.
• Evaluate each condition logically to ensure no contradictions exist.
• Rule out impossibilities thoughtfully and thoroughly.

This mathematical process helps ensure accurate problem solving, allowing you to make well-supported conclusions. This is a key technique for successfully tackling geometry problems.

Solving Geometry Problems

Solving geometry problems often requires systematic processes and proper use of theorems like the Triangle Inequality Theorem. This becomes crucial when you need to find out relationships among different dimensions within geometric shapes. Understanding and applying these concepts can be deeply insightful:

• Identify the sides of a triangle and factor in their relationships.
• Apply relevant geometric principles such as the Triangle Inequality Theorem.
• Use logic to confirm or eliminate possible solutions.

By delving deep into geometry problems, you learn to connect various mathematical concepts and find solutions efficiently. This practical application of learning is essential, and sometimes challenging exercises help reinforce these fundamental concepts.