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Chapter 7: Problem 20

Determine the number of triangles with the given parts. $$a=3, b=15, c=16$$

Short Answer

Expert verified
1 triangle

Step by step solution

01

Identify the Type of Problem

Given the sides of a triangle, it is important to check if the triangle inequality holds. The triangle inequality theorem states that the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
02

Apply the Triangle Inequality Theorem

Verify if the given sides satisfy the triangle inequality theorem. For sides a, b, and c:a + b > ca + c > bb + c > aSubstituting the given values a = 3, b = 15, and c = 16:
03

Evaluate Each Inequality

1. a + b > c: 3 + 15 > 1618 > 16 (True)2. a + c > b: 3 + 16 > 1519 > 15 (True)3. b + c > a:15 + 16 > 331 > 3 (True)
04

Conclusion Based on Inequality Evaluation

Since all three inequalities are satisfied, the given side lengths can form a triangle. Thus, there is one triangle that can be formed with the sides a = 3, b = 15, and c = 16.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Triangle Formation
Understanding triangle formation is crucial in solving many geometry problems. A triangle is defined by three sides and three angles. The sides must satisfy the triangle inequality theorem for a triangle to be formed.
In our exercise, we were given three side lengths: a = 3, b = 15, and c = 16.
To determine if these sides can form a triangle, we check:
  • The sum of any two sides must be greater than or equal to the third side.
Verifying this ensures no side is too long to exclude the possibility of forming a triangle.
Inequalities
The concept of inequalities plays a significant role in geometry, particularly in determining triangle formation. The triangle inequality theorem outlines the critical relationships between the sides of a triangle. Here's how you apply it:
For sides a, b, and c:
  • \( a + b > c \)
  • \( a + c > b \)
  • \( b + c > a \)
Given our side lengths: a = 3, b = 15, and c = 16, we checked:
  • 3 + 15 > 16 (True)
  • 3 + 16 > 15 (True)
  • 15 + 16 > 3 (True)
Since all conditions are met, the inequalities confirm that these sides can form a triangle.
Geometry Problem-Solving
Solving geometry problems requires a clear and methodical approach. Here鈥檚 a strategy you can use:
1. **Understand the problem:** Identify what needs to be proven or found, such as verifying if given sides can form a triangle.
2. **Apply relevant theorems:** Use known theorems and properties, like the triangle inequality theorem, to verify conditions.
3. **Do the math:** Substitute values and verify each condition step-by-step.
For our exercise, these steps ensured we systematically confirmed that the sides a = 3, b = 15, and c = 16 can indeed form a triangle. Following processes like this helps ensure accuracy and understanding in geometry problem-solving.

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