Marlenas Obtuse Triangle Puzzle Determining Shortest Straw Length

Introduction: The Triangle Inequality and Obtuse Triangles

The fascinating world of geometry often presents us with intriguing puzzles, and this one is no exception. In this article, we will delve into a problem involving Marlena, her three straws of varying lengths, and the challenge of forming an obtuse triangle. To solve this puzzle, we need to leverage our understanding of the triangle inequality theorem and the properties of obtuse 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 governs the formation of triangles. An obtuse triangle, on the other hand, is a triangle in which one of the angles is greater than 90 degrees. This characteristic has a direct impact on the relationship between the sides of the triangle, which we will explore further. Our goal is to determine the possible lengths of the shortest straw, given the lengths of the other two straws and the fact that the triangle formed is obtuse. This requires a careful application of the triangle inequality and an understanding of how side lengths relate to angles in a triangle. Let's embark on this geometric journey and unravel the mystery of the shortest straw.

Setting Up the Problem: Marlena's Straws and the Obtuse Condition

Marlena has three straws, each with a different length. We are given the lengths of two straws, but the length of the shortest straw remains a mystery. Let's denote the lengths of the two known straws as 'a' and 'b', and the length of the shortest straw as 'c'. We know that c < a and c < b. The crucial piece of information is that when Marlena forms a triangle using all three straws, the resulting triangle is obtuse. This condition places a significant constraint on the possible lengths of the shortest straw. To solve this, we need to translate this geometric condition into a mathematical inequality. Recall that in an obtuse triangle, the square of the longest side is greater than the sum of the squares of the other two sides. This is a direct consequence of the Pythagorean theorem, which applies to right triangles. In the case of an obtuse triangle, the angle opposite the longest side is greater than 90 degrees, leading to this inequality. We will use this relationship, along with the triangle inequality, to determine the possible values of 'c'. The challenge lies in combining these two concepts effectively to narrow down the range of possible lengths for the shortest straw. By carefully considering the relationships between the sides of the triangle and the obtuse angle condition, we can unlock the solution to this geometric puzzle. The following sections will walk you through the steps of setting up the inequalities and solving for the unknown length.

Applying the Triangle Inequality Theorem

To begin our analysis, we must first ensure that the three straws can indeed form a triangle. This is where the triangle inequality theorem comes into play. As mentioned earlier, this 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. In Marlena's case, this translates into three inequalities: a + b > c, a + c > b, and b + c > a. These inequalities are the foundation of our solution, as they define the basic requirements for the straws to form a triangle. If any of these inequalities are not satisfied, then no triangle can be formed, regardless of whether it is obtuse or acute. The first inequality, a + b > c, simply ensures that the two longer straws are long enough to meet and form a triangle with the shortest straw. The second inequality, a + c > b, states that the sum of one of the longer straws and the shortest straw must be greater than the other longer straw. Similarly, the third inequality, b + c > a, states that the sum of the other longer straw and the shortest straw must be greater than the first longer straw. These inequalities might seem straightforward, but they are crucial in defining the lower bound for the possible lengths of the shortest straw. They prevent the shortest straw from being too short, ensuring that it can connect the other two straws to form a closed figure. By carefully considering these inequalities, we can establish a range of possible values for 'c' that satisfy the basic condition of triangle formation. This is the first step in our journey to determine the specific lengths that result in an obtuse triangle.

The Obtuse Angle Condition: Side Length Relationships

Now that we've established the basic conditions for forming a triangle, let's incorporate the information about the obtuse angle. Recall that in an obtuse triangle, the square of the length of the longest side is greater than the sum of the squares of the lengths of the other two sides. This relationship stems from the Law of Cosines, which generalizes the Pythagorean theorem to all triangles. In our case, we need to identify the longest side and apply this obtuse angle condition. Let's assume, without loss of generality, that 'a' is the longest straw (i.e., a > b and a > c). Then, the obtuse angle condition can be expressed as a² > b² + c². This inequality is the key to unlocking the solution. It tells us that the square of the length of the longest straw must be significantly larger than the combined squares of the other two straws for the triangle to be obtuse. If the square of the longest side were equal to the sum of the squares of the other two sides, we would have a right triangle. If it were smaller, we would have an acute triangle. The