Problem 4 Check if it is possible for a tr... [FREE SOLUTION] (2024)

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Chapter 2: Problem 4

Check if it is possible for a triangle to have the given parts. \(a=5, b=7, c=10, A=27.7^{\circ}, B=40.5^{\circ}, C=111.8\)

Short Answer

Expert verified

Yes, it is possible for a triangle to have these given parts.

Step by step solution

01

- Verify Angle Sum

Check if the sum of the given angles is equal to 180 degrees. Calculate: \(A + B + C = 27.7^{\text{◦}} + 40.5^{\text{◦}} + 111.8^{\text{◦}} = 180^{\text[]{◦}}\)

02

- Check Triangle Inequality

Verify that the given side lengths satisfy the triangle inequality theorem. This theorem states that the sum of any two sides must be greater than the third side. Check the following inequalities: \(a + b > c \rightarrow 5 + 7 > 10 → 12 > 10\) \(a + c > b \rightarrow 5 + 10 > 7 → 15 > 7\) \(b + c > a \rightarrow 7 + 10 > 5 → 17 > 5\)

03

- Short Answer

The angles sum to 180 degrees and the sides satisfy the triangle inequality theorem, so the given parts can form a triangle.

Key Concepts

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

Angle Sum Property

In any triangle, the sum of the interior angles is always equal to 180 degrees. This is known as the 'Angle Sum Property'.
The given problem asks us to check if a triangle with angles 27.7°, 40.5°, and 111.8° can exist. According to the property:
\[ A + B + C = 180^\text{∘} \]
By substituting the given values, we get:
\[ 27.7^\text{∘} + 40.5^\text{∘} + 111.8^\text{∘} = 180^\text{∘} \]
The sum is indeed 180 degrees. Therefore, based on this property alone, the angles can form a triangle.
This property is fundamental because it applies to all triangles, regardless of their type (scalene, isosceles, or equilateral).

Triangle Inequality Theorem

The Triangle Inequality Theorem states that for three line segments to create a triangle, the sum of the lengths of any two sides must be greater than the length of the third side.
In our problem, the side lengths given are 5, 7, and 10. We need to check three inequalities:

  • \(a + b > c\)
  • \(a + c > b\)
  • \(b + c > a\)

For our specific values, this becomes:

  • \(5 + 7 > 10\; (12 > 10)\)
  • \(5 + 10 > 7\; (15 > 7)\)
  • \(7 + 10 > 5\; (17 > 5)\)

All three inequalities are satisfied, indicating that the given sides can indeed form a triangle.
This theorem ensures that the sides are long enough to connect and form a closed shape.
It's essential to verify this, otherwise we might think it's possible to form a triangle when it isn't.

Triangle Formation Conditions

For a set of segments and angles to form a triangle, specific conditions must be met. These conditions ensure the consistency and validity of the shape.
1. The sum of the internal angles must be exactly 180 degrees (Angle Sum Property).
2. The lengths of the sides must satisfy the Triangle Inequality Theorem, i.e., the sum of any two sides should be greater than the third side.
In our exercise, both conditions are met:

  • Angle sum: \(27.7^\text{∘} + 40.5^\text{∘} + 111.8^\text{∘} = 180^\text{∘}\)
  • Side lengths: \(5, 7, 10\)

By verifying these conditions, we confirm that a triangle with the given parts can indeed be formed.
Understanding these conditions is crucial for accurately determining the possibility of triangle formation and avoiding common mistakes in geometric problems.

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Problem 4 Check if it is possible for a tr... [FREE SOLUTION] (3)

Most popular questions from this chapter

Solve the triangle \(\triangle A B C\). \(a=5, A=47^{\circ}, b=9\)Solve the triangle \(\triangle A B C\). \(a=12, A=94^{\circ}, b=15\)Two trains leave the same train station at the same time, moving alongstraight tracks that form a \(35^{\circ}\) angle. If one train travels at anaverage speed of \(100 \mathrm{mi} / \mathrm{hr}\) and the other at an averagespeed of \(90 \mathrm{mi} / \mathrm{hr}\), how far apart are the trains afterhalf an hour?Use the Law of Cosines to show that for any triangle \(\triangle A B C,c^{2}a^{2}+b^{2}\) if \(C\) is obtuse, and\(c^{2}=a^{2}+b^{2}\) if \(C\) is a right angle.Solve the triangle \(\triangle A B C\). \(A=30^{\circ}, b=4, c=6\)
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Problem 4 Check if it is possible for a tr... [FREE SOLUTION] (2024)
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