Prove Points A(a, 0), B(-a, 0), C(0, A √3) Form Equilateral Triangle
In the fascinating realm of geometry, an equilateral triangle stands out as a figure of perfect harmony, boasting three equal sides and three equal angles, each measuring a precise 60 degrees. This exploration delves into the captivating problem of demonstrating that the points A(a, 0), B(-a, 0), and C(0, a √3) coalesce to form such an equilateral marvel. To embark on this mathematical journey, we shall harness the power of the distance formula, a fundamental tool that unveils the distance between two points nestled within a coordinate plane. This formula, rooted in the Pythagorean theorem, allows us to quantify the lengths of the line segments connecting our trio of points, thereby unveiling the triangle's inherent geometric properties.
The Distance Formula: A Cornerstone of Geometric Discovery
At the heart of our endeavor lies the distance formula, a mathematical expression that elegantly captures the Euclidean distance between two points in a coordinate plane. Let's consider two points, P(x₁, y₁) and Q(x₂, y₂). The distance, d, between these points is meticulously calculated as follows:
d = √((x₂ - x₁)² + (y₂ - y₁)²)
This formula, derived from the illustrious Pythagorean theorem, serves as our trusty compass, guiding us through the geometric landscape. It empowers us to precisely determine the lengths of the line segments that constitute our triangle, thereby paving the way for unraveling its equilateral nature. Understanding and applying this formula is paramount to our exploration, as it forms the bedrock upon which our geometric deductions are built.
Calculating the Distances: Unveiling the Triangle's Dimensions
Armed with the distance formula, we now embark on a meticulous calculation of the distances between our points A(a, 0), B(-a, 0), and C(0, a √3). This step is pivotal, as it allows us to quantify the lengths of the triangle's sides, providing the numerical evidence needed to ascertain its equilateral nature. Let's delve into the calculations:
Distance Between A and B:
Applying the distance formula to points A(a, 0) and B(-a, 0), we have:
AB = √((-a - a)² + (0 - 0)²) = √((-2a)²) = √(4a²) = 2|a|
This calculation reveals the length of side AB to be 2|a|, where |a| denotes the absolute value of a. The absolute value ensures that the distance is always a non-negative quantity, regardless of the sign of 'a'.
Distance Between B and C:
Next, we compute the distance between points B(-a, 0) and C(0, a √3):
BC = √((0 - (-a))² + (a√3 - 0)²) = √(a² + (a√3)²) = √(a² + 3a²) = √(4a²) = 2|a|
The length of side BC also emerges as 2|a|, further hinting at the triangle's equilateral nature. The consistency in the side lengths is a crucial piece of evidence in our geometric puzzle.
Distance Between C and A:
Finally, we determine the distance between points C(0, a √3) and A(a, 0):
CA = √((a - 0)² + (0 - a√3)²) = √(a² + (-a√3)²) = √(a² + 3a²) = √(4a²) = 2|a|
Remarkably, the length of side CA also equals 2|a|, completing the trifecta of equal side lengths. This conclusive calculation solidifies our suspicion that the triangle formed by points A, B, and C is indeed equilateral.
Verifying the Equilateral Nature: A Symphony of Equal Sides
Our meticulous calculations have unveiled a profound truth: the distances between the points A(a, 0), B(-a, 0), and C(0, a √3) are all equal. Specifically, we have established that:
AB = BC = CA = 2|a|
This harmonious equality of side lengths is the defining characteristic of an equilateral triangle. By definition, an equilateral triangle is a polygon with three sides of the same length. Our calculations have demonstrably satisfied this condition, providing irrefutable evidence that the triangle formed by our points is indeed equilateral. This verification marks a triumphant milestone in our geometric exploration.
Conclusion: A Geometric Harmony Unveiled
In this comprehensive exploration, we have successfully demonstrated that the points A(a, 0), B(-a, 0), and C(0, a √3) gracefully form an equilateral triangle. Our journey began with the distance formula, a powerful tool that allowed us to precisely quantify the distances between the points. Through meticulous calculations, we unveiled the remarkable fact that all three sides of the triangle possess the same length, a defining characteristic of equilateral triangles. This harmonious equality of sides serves as the cornerstone of our conclusion, solidifying the geometric truth that we set out to uncover. The elegance and precision of mathematics have once again illuminated the inherent beauty and order within the world of geometric forms.