An Acute Triangle Has Sides Of 10 Cm And 16 Cm. What Range Of Lengths Is Possible For The Third Side?

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Determining the possible range of values for the third side of an acute triangle when two sides are known is a fascinating problem in geometry. This exploration delves into the acute triangle with sides measuring 10 cm and 16 cm, seeking to define the constraints on the unknown third side. To solve this, we'll use the triangle inequality theorem and the properties of acute triangles. The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This fundamental principle helps us establish the initial bounds for the possible values of the unknown side. Let's denote the unknown side as x. According to the triangle inequality theorem, we have three inequalities: 10 + 16 > x, 10 + x > 16, and 16 + x > 10. Solving these inequalities gives us a preliminary range for x. However, the condition that the triangle is acute adds another layer of complexity. In an acute triangle, all angles are less than 90 degrees. This translates into a condition on the sides: the square of the longest side must be less than the sum of the squares of the other two sides. This condition further refines the possible values of x, ensuring that the resulting triangle is indeed acute. We need to consider two cases: when 16 cm is the longest side and when x is the longest side. By applying the acute triangle condition to both cases, we derive additional inequalities that constrain the range of x. Combining the results from the triangle inequality theorem and the acute triangle condition, we arrive at the final range of possible values for the third side. This range represents the set of all lengths that satisfy both the triangle inequality and the acute triangle property. The problem highlights the interplay between different geometric principles and showcases how they can be used to solve complex problems. Understanding these concepts is crucial for mastering geometry and its applications in various fields.

Triangle Inequality Theorem

To understand the range of the third side, we must first recall the triangle inequality theorem. This theorem is a cornerstone of triangle geometry and dictates the fundamental relationship between the sides of any triangle. The theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This principle ensures that the sides can actually “connect” to form a closed figure, a necessary condition for a triangle to exist. Without this condition, the two shorter sides would not be long enough to meet and create a triangle. This theorem provides a basic framework for determining the possible lengths of the third side when the lengths of the other two sides are known. We can apply this theorem to any triangle, regardless of its angles or specific dimensions. The triangle inequality theorem is not just an abstract concept; it has practical applications in various fields, such as engineering, architecture, and surveying. For instance, engineers use this theorem to ensure the stability of structures, while architects use it to design buildings with sound structural integrity. Surveyors rely on the theorem to accurately measure distances and angles in the field. This theorem can be visualized by imagining three line segments of different lengths. If the sum of the two shorter segments is less than or equal to the length of the longest segment, it is impossible to form a triangle. The two shorter segments would simply lie flat along the longest segment, without ever meeting to close the shape. Only when the sum of the two shorter segments is strictly greater than the longest segment can a triangle be formed. This simple yet powerful concept is the essence of the triangle inequality theorem. The theorem allows us to establish initial boundaries for the possible values of the third side. However, it only provides a necessary condition for a triangle to exist; it does not guarantee that the triangle will have specific properties, such as being acute, obtuse, or right-angled. For such specific properties, we need to consider additional conditions, such as the Pythagorean theorem and its extensions.

Acute Triangle Condition

Beyond the basic requirement of forming a triangle, the problem specifies that the triangle is acute. This adds another layer of constraint on the possible values of the third side. In an acute triangle, all three angles are less than 90 degrees. This condition translates into a specific relationship between the squares of the side lengths. Let's denote the sides of the triangle as a, b, and c, where c is the longest side. For an acute triangle, the following condition must hold: c² < a² + b². This inequality is a direct consequence of the cosine rule, which relates the sides and angles of any triangle. When all angles are less than 90 degrees, the cosine of each angle is positive, leading to the inequality above. This acute triangle condition is crucial for narrowing down the possible values of the third side. It ensures that the triangle not only exists but also satisfies the acute angle requirement. Without this condition, we might find solutions that form valid triangles but are not acute. The acute triangle condition is a specific case of a more general relationship between side lengths and angles in a triangle. For an obtuse triangle, where one angle is greater than 90 degrees, the inequality becomes c² > a² + b². For a right-angled triangle, where one angle is exactly 90 degrees, the equality c² = a² + b² holds, which is the Pythagorean theorem. The acute triangle condition is not just a theoretical concept; it has practical applications in various fields, such as structural engineering and computer graphics. Engineers use this condition to design structures that are stable and can withstand external forces. In computer graphics, the condition is used to ensure that triangles used in 3D models are properly formed and do not have any distorted angles. Understanding the acute triangle condition requires a solid grasp of the relationship between sides and angles in a triangle. It also highlights the importance of considering specific properties of triangles when solving geometric problems. By applying this condition in conjunction with the triangle inequality theorem, we can precisely determine the range of possible values for the third side of an acute triangle. This ensures that our solution satisfies all the given constraints and accurately represents the geometry of the problem.

Applying the Theorems to Find the Range

Now, let's apply both the triangle inequality theorem and the acute triangle condition to our specific problem. We have a triangle with sides 10 cm and 16 cm, and we want to find the range of possible values for the third side, x. First, we apply the triangle inequality theorem. We have three inequalities:

  1. 10 + 16 > x
  2. 10 + x > 16
  3. 16 + x > 10

Solving these inequalities:

  1. 26 > x or x < 26
  2. x > 6
  3. x > -6 (This inequality is always true since x must be positive)

Combining these results, we get a preliminary range for x: 6 < x < 26. This range tells us that the third side must be longer than 6 cm and shorter than 26 cm to form a triangle. However, we also need to consider the acute triangle condition. We have two cases to consider:

Case 1: 16 cm is the longest side

In this case, we have 16² < 10² + x², which simplifies to 256 < 100 + x². This gives us x² > 156, so x > √156 ≈ 12.49. Thus, when 16 cm is the longest side, the third side must be greater than approximately 12.49 cm.

Case 2: x is the longest side

In this case, we have x² < 10² + 16², which simplifies to x² < 100 + 256, or x² < 356. This gives us x < √356 ≈ 18.87. Thus, when x is the longest side, it must be less than approximately 18.87 cm.

Combining the results from both cases and the triangle inequality theorem, we get the final range for x: 12.49 < x < 18.87. This range represents the possible lengths of the third side that satisfy both the triangle inequality and the acute triangle condition. Any value of x within this range will result in an acute triangle with sides 10 cm and 16 cm. The process of applying these theorems demonstrates the power of geometric principles in solving practical problems. By combining different theorems and conditions, we can precisely determine the possible solutions and gain a deeper understanding of the underlying geometry. This approach is applicable to a wide range of geometric problems and is a fundamental skill for anyone studying mathematics or related fields.

Final Range and Conclusion

In conclusion, by applying the triangle inequality theorem and the acute triangle condition, we have successfully determined the range of possible values for the third side of the acute triangle. The triangle inequality theorem gave us the preliminary range of 6 < x < 26. The acute triangle condition, considered in two cases, further narrowed this range. When 16 cm was the longest side, we found that x > √156 ≈ 12.49 cm. When x was the longest side, we found that x < √356 ≈ 18.87 cm. Combining all these conditions, we arrive at the final range: 12.49 < x < 18.87. This range represents the set of all possible lengths for the third side that will form an acute triangle with sides 10 cm and 16 cm. Any value of x within this range satisfies both the triangle inequality and the acute triangle condition. This problem demonstrates the importance of considering all relevant geometric principles when solving a problem. The triangle inequality theorem is a fundamental condition for the existence of a triangle, while the acute triangle condition ensures that the triangle has specific properties related to its angles. By applying both theorems, we were able to precisely determine the possible solutions. The process of solving this problem highlights the interplay between different geometric concepts and the importance of a systematic approach. Understanding these principles is crucial for mastering geometry and its applications in various fields. The range of possible values for the third side is not just a numerical answer; it represents a geometric constraint that must be satisfied for the triangle to exist and have the specified properties. This understanding is essential for solving more complex geometric problems and for applying geometric principles in real-world scenarios. The ability to combine different theorems and conditions to solve a problem is a key skill in mathematics and is applicable to many other areas of science and engineering.

The correct answer is the range 12.49 < x < 18.87. This signifies that the third side must be longer than 12.49 cm and shorter than 18.87 cm to form an acute triangle with the given sides.