Finding The Length Of A² In An Isosceles Triangle A Step By Step Solution
In the realm of geometry, isosceles triangles hold a special allure, characterized by their two equal sides and two equal angles. Delving into the properties of these triangles often leads to fascinating mathematical explorations. In this article, we embark on a journey to unravel the length of a² in a specific isosceles triangle ABC, where angle A measures π/4 and sides b and c are both equal to 3. Our quest will involve applying fundamental trigonometric principles and algebraic manipulations to arrive at the solution.
Defining the Isosceles Triangle ABC
To begin, let's meticulously define the isosceles triangle ABC that forms the crux of our investigation. We are given that angle A is equal to π/4, which translates to 45 degrees. This immediately establishes triangle ABC as an acute isosceles triangle, as all its angles are less than 90 degrees. Furthermore, we know that sides b and c are both equal to 3, signifying that these two sides are the equal sides of the isosceles triangle. The side opposite angle A, denoted as side a, is the side we aim to determine.
Visualizing the Triangle
Before we plunge into calculations, let's conjure a mental image of the isosceles triangle ABC. Picture a triangle with two sides of equal length (b = c = 3) converging at a vertex where the angle between them (angle A) is 45 degrees. The remaining side, side a, stretches across from the opposite vertex, completing the triangular form. This visual representation will aid us in grasping the spatial relationships and applying the appropriate trigonometric laws.
Laying the Groundwork: Key Geometric Principles
Our pursuit of the length of a² hinges on two fundamental geometric principles: the Law of Cosines and the properties of isosceles triangles. Let's briefly revisit these principles to ensure a solid foundation for our calculations.
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Law of Cosines: This cornerstone of trigonometry provides a relationship between the sides and angles of any triangle. For triangle ABC, the Law of Cosines states:
a² = b² + c² - 2bc * cos(A)This equation will serve as our primary tool in determining the length of a².
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Properties of Isosceles Triangles: Isosceles triangles possess unique characteristics that simplify our calculations. Specifically, we know that the base angles (angles B and C) of an isosceles triangle are equal. Furthermore, the sum of all angles in a triangle is always 180 degrees (or π radians). These properties will help us deduce the measures of angles B and C.
Embarking on the Calculation Journey
With the isosceles triangle ABC defined and the necessary geometric principles in hand, we are now poised to embark on the calculation journey. Our primary objective is to determine the length of a², which we will achieve by strategically applying the Law of Cosines and leveraging the properties of isosceles triangles.
Step 1: Invoking the Law of Cosines
Our first step involves directly applying the Law of Cosines to triangle ABC. We know the lengths of sides b and c (both equal to 3) and the measure of angle A (π/4 radians). Plugging these values into the Law of Cosines equation, we get:
a² = 3² + 3² - 2 * 3 * 3 * cos(π/4)
This equation now expresses a² in terms of known quantities, bringing us closer to our goal.
Step 2: Evaluating the Cosine Term
To proceed further, we need to evaluate the cosine term, cos(π/4). Recall that π/4 radians corresponds to 45 degrees. The cosine of 45 degrees is a well-known trigonometric value, equal to √2 / 2. Substituting this value into our equation, we get:
a² = 3² + 3² - 2 * 3 * 3 * (√2 / 2)
This substitution simplifies the equation, paving the way for further algebraic manipulations.
Step 3: Simplifying the Expression
Now, let's simplify the expression by performing the arithmetic operations. We have:
a² = 9 + 9 - 9√2
Combining the constants, we arrive at:
a² = 18 - 9√2
Step 4: Factoring for Elegance
To present the result in a more elegant and insightful form, we can factor out a common factor of 9 from the expression:
a² = 9(2 - √2)
This factored form reveals the structure of the solution and allows for easy comparison with the given options.
The Grand Finale: Identifying the Solution
After our meticulous calculations, we have arrived at the solution for the length of a²:
a² = 9(2 - √2)
This result perfectly matches option C in the given choices. Therefore, the length of a² in the isosceles triangle ABC is 3²(2 - √2).
Reflecting on the Solution
Our journey to determine the length of a² has been a rewarding exercise in applying geometric principles and algebraic techniques. We successfully utilized the Law of Cosines and the properties of isosceles triangles to arrive at the solution. The factored form of the answer, 9(2 - √2), provides a concise and insightful representation of the relationship between the sides and angles of the triangle. This exploration reinforces the power of mathematical tools in unraveling geometric mysteries.
Further Explorations
As we conclude our investigation, it's worth pondering further explorations related to this isosceles triangle. For instance, we could calculate the lengths of the altitudes, the area of the triangle, or the circumradius and inradius. These explorations would provide a deeper understanding of the triangle's properties and its place within the broader landscape of geometry. The possibilities are vast, and the journey of mathematical discovery never truly ends.
Conclusion
In conclusion, we have successfully determined the length of a² in the isosceles triangle ABC, where angle A is π/4 and sides b and c are both equal to 3. Through the strategic application of the Law of Cosines and the properties of isosceles triangles, we arrived at the solution a² = 3²(2 - √2). This exploration underscores the elegance and power of mathematical principles in solving geometric problems. The world of geometry is rich with such challenges, and each solution serves as a stepping stone to further discoveries.
