5-4 Additional Practice Inequalities In One Triangle

5 + 4: Mastering Additional Practice Inequalities in One Triangle

Understanding triangle inequalities is crucial for mastering geometry. While the basic triangle inequality theorem – the sum of any two sides must be greater than the third – forms the foundation, several additional inequalities build upon this principle, providing a more nuanced understanding of triangle relationships. This comprehensive guide delves into five fundamental inequalities, followed by four additional, more advanced ones, offering ample practice problems and explanations to solidify your grasp of this essential geometric concept.

The Five Fundamental Triangle Inequalities

These five inequalities serve as the bedrock upon which more complex triangle inequality problems are built. Understanding them thoroughly is paramount before tackling the more advanced concepts.

1. The Triangle Inequality Theorem (TIT): The Foundation

This is the cornerstone of all triangle inequalities. It states: The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

• a + b > c
• a + c > b
• b + c > a

Where 'a', 'b', and 'c' represent the lengths of the sides of the triangle. If any of these inequalities are not true, then a triangle cannot be formed with those side lengths.

Practice Problem 1: Can a triangle be formed with sides of length 5, 7, and 13?

Solution: 5 + 7 = 12 < 13. Since the sum of two sides is less than the third, a triangle cannot be formed.

2. The Relationship Between Sides and Angles: The Larger Angle Opposite the Larger Side

This inequality establishes a direct correlation between the lengths of the sides and the measures of their opposite angles: In any triangle, the largest angle is opposite the longest side, and the smallest angle is opposite the shortest side.

Practice Problem 2: Triangle ABC has sides AB = 6, BC = 8, and AC = 10. Which angle is the largest?

Solution: The longest side is AC (10), therefore, the largest angle is ∠B (opposite AC).

3. The Angle Bisector Theorem Inequality: Dividing the Triangle

The angle bisector of a triangle divides the opposite side into segments proportional to the adjacent sides. While not strictly an inequality in the same vein as the others, it leads to inequalities when comparing segment lengths: If AD is the angle bisector of ∠A in triangle ABC, then AB/AC = BD/DC. This implies inequalities related to the relative lengths of BD and DC based on the lengths of AB and AC.

Practice Problem 3: In triangle ABC, AB = 8, AC = 6, and AD bisects ∠A. Which segment, BD or DC, is longer?

Solution: Since AB > AC, then BD > DC.

4. The Exterior Angle Inequality: Angles Outside the Triangle

The measure of an exterior angle of a triangle is greater than the measure of either of its remote interior angles. This leads to various inequalities related to the angles' magnitudes.

Practice Problem 4: In triangle ABC, ∠ACD is an exterior angle. If ∠BAC = 50° and ∠ABC = 70°, what can be said about the magnitude of ∠ACD?

Solution: ∠ACD > 50° and ∠ACD > 70°. Therefore, ∠ACD > 70°.

5. The Inequality of Medians: Connecting to the Centroid

The sum of the lengths of any two medians of a triangle is greater than the length of the third median. This is analogous to the basic triangle inequality theorem, but applied to medians instead of sides.

Practice Problem 5: In triangle ABC, medians AD, BE, and CF are drawn. If AD = 6 and BE = 8, what can be said about the length of CF?

Solution: CF < 14 (6 + 8) and CF > 2 (|8-6|).

Four Advanced Triangle Inequalities

These inequalities build upon the foundational ones, requiring a deeper understanding of geometric relationships and proof techniques.

6. The Erdős–Mordell Inequality: Distances from a Point Inside

This theorem states that for any point P inside a triangle ABC, the sum of the distances from P to the sides is greater than or equal to twice the distance from P to the circumcenter O. More formally: PA + PB + PC ≥ 2(d + r), where PA, PB, PC are distances from P to vertices A, B, C respectively and d is the distance of P from the centroid and r is the inradius. This inequality has a beautiful elegance and showcases the powerful relationships within a triangle.

7. The Weitzenböck Inequality: Squares of Distances

For a triangle with sides a, b, c, and its area denoted by A, the following inequality holds: a² + b² + c² ≥ 4√3A. This inequality connects the side lengths and the area of the triangle, highlighting the interrelation of these metrics.

8. Ivic's Inequality: Refinement of the Triangle Inequality Theorem

Ivić's inequality provides a sharper bound than the basic triangle inequality. For a triangle with sides a, b, and c, it states: (a + b - c)² ≤ 4ab(1 - cos C). This inequality provides a tighter constraint on the relationship between the side lengths and the cosine of the angle between them.

9. The Ptolemy's Inequality: Cyclic Quadrilaterals and Triangles

While not strictly an inequality within a triangle, Ptolemy's inequality relates to cyclic quadrilaterals, which are quadrilaterals that can be inscribed in a circle. This inequality can be applied to triangles as well, providing a relationship between the sides and the diagonals of a quadrilateral which can be viewed as a degenerate form of a cyclic quadrilateral containing a triangle. The inequality states: ab + cd ≥ ef, where a, b, c, d are the side lengths of the quadrilateral, and e, f are the lengths of the diagonals. This connection highlights the interrelationship between geometric shapes and their properties.

Practice Problems: Putting it all Together

Here are some more complex practice problems that integrate multiple inequalities:

Problem 1: A triangle has sides of length 7, x, and 11. Find the range of possible values for x.

Problem 2: In triangle ABC, AB = 5, BC = 7, and AC = 8. If the angle bisector of angle B intersects AC at D, find the possible ranges of lengths for AD and DC.

Problem 3: Triangle ABC has sides a, b, and c. Show that a² + b² + c² < 2(ab + bc + ca).

Problem 4: If a triangle has sides a, b, and c, show using Ivić's inequality that the triangle inequality theorem holds.

Problem 5: Given a triangle with sides 5, 12, and 13, calculate the area and verify the Weitzenböck inequality.

Solving these problems will require you to apply multiple inequalities and understand the interplay between side lengths, angles, medians, and areas within a triangle. Remember to carefully consider each inequality's implications and how they relate to each other. Thorough understanding of these inequalities will significantly enhance your geometry skills and problem-solving capabilities. Good luck and happy problem-solving!