Prove Right-Angled Isosceles Triangle With Slopes

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Introduction

In the realm of coordinate geometry, triangles hold a fundamental position, with their properties and classifications forming the bedrock of numerous geometric concepts. Among these, the right-angled isosceles triangle stands out due to its unique combination of features: a right angle and two sides of equal length. This article delves into the method of proving that three given points form a right-angled isosceles triangle by utilizing the concept of slopes. Slopes, a measure of the steepness and direction of a line, provide a powerful tool for analyzing the relationships between lines and, consequently, the angles formed by them. Our specific task is to demonstrate, using slopes, that the points (1,1), (2,3), and (-1,2) indeed constitute such a triangle. This exploration will not only reinforce the understanding of slope calculations but also illuminate how slopes can be employed to deduce crucial geometric properties of triangles. Furthermore, this approach provides a rigorous analytical method, complementing traditional geometric proofs and offering a deeper insight into the interplay between algebra and geometry.

Understanding Slopes and Their Significance

Before we plunge into the proof, it is crucial to solidify our understanding of slopes. The slope of a line, mathematically defined as the "rise over run," quantifies the steepness and direction of a line segment. It is calculated as the change in the y-coordinate divided by the change in the x-coordinate between any two points on the line. Formally, if we have two points (x1, y1) and (x2, y2), the slope (m) of the line joining them is given by the formula:

m = (y2 - y1) / (x2 - x1)

Slopes play a pivotal role in determining the relationships between lines. Parallel lines, for instance, possess equal slopes, indicating that they share the same inclination and will never intersect. Conversely, perpendicular lines exhibit a unique relationship: the product of their slopes is -1. This property stems from the fact that perpendicular lines intersect at a right angle, forming a cornerstone of geometric analysis. In our context of triangles, slopes become invaluable in identifying right angles. If two sides of a triangle have slopes whose product equals -1, we can definitively conclude that those sides are perpendicular, and the triangle contains a right angle. This connection between slopes and perpendicularity is what we will leverage to prove the right-angled nature of our triangle.

Calculating Slopes for the Given Points

Now, let's apply our understanding of slopes to the specific points in question: (1,1), (2,3), and (-1,2). We'll denote these points as A(1,1), B(2,3), and C(-1,2), respectively, for clarity. Our next step is to calculate the slopes of the lines formed by connecting these points pairwise. This will give us a quantitative measure of the inclination of each side of the triangle.

Slope of AB

To find the slope of line segment AB, we apply the slope formula using points A(1,1) and B(2,3):

m_AB = (3 - 1) / (2 - 1) = 2 / 1 = 2

Thus, the slope of AB is 2. This indicates a positive slope, meaning the line segment rises as it moves from left to right.

Slope of BC

Next, we calculate the slope of line segment BC using points B(2,3) and C(-1,2):

m_BC = (2 - 3) / (-1 - 2) = -1 / -3 = 1/3

The slope of BC is 1/3, also positive, but less steep than AB.

Slope of AC

Finally, we determine the slope of line segment AC using points A(1,1) and C(-1,2):

m_AC = (2 - 1) / (-1 - 1) = 1 / -2 = -1/2

The slope of AC is -1/2, a negative slope, indicating the line segment falls as it moves from left to right.

With the slopes of all three sides calculated, we are now equipped to analyze the relationships between these lines and ascertain the angles within the triangle.

Proving the Right Angle

Having computed the slopes of the sides AB, BC, and AC, we now focus on demonstrating that one of the angles in the triangle is a right angle. Recall that two lines are perpendicular if and only if the product of their slopes is -1. This is the key criterion we will use to identify a right angle within the triangle. We will systematically examine the product of the slopes of each pair of sides to see if any pair satisfies this condition.

Checking for Perpendicularity

  • AB and BC: The product of their slopes is 2 * (1/3) = 2/3, which is not equal to -1. Therefore, AB and BC are not perpendicular.
  • AB and AC: The product of their slopes is 2 * (-1/2) = -1. This result confirms that AB and AC are indeed perpendicular, implying that the angle formed at vertex A is a right angle.
  • BC and AC: The product of their slopes is (1/3) * (-1/2) = -1/6, which is not equal to -1. Thus, BC and AC are not perpendicular.

The crucial finding here is that the product of the slopes of AB and AC is -1. This definitively proves that lines AB and AC are perpendicular. Consequently, the angle formed at vertex A, ∠BAC, is a right angle. This establishes the first characteristic of a right-angled triangle.

Proving the Isosceles Nature

Now that we've established the presence of a right angle, the next step is to demonstrate that the triangle is also isosceles, meaning it has two sides of equal length. To achieve this, we will calculate the lengths of the sides AB, AC, and BC using the distance formula. The distance formula, derived from the Pythagorean theorem, provides a method for determining the distance between two points in a coordinate plane. For points (x1, y1) and (x2, y2), the distance (d) between them is given by:

d = √[(x2 - x1)² + (y2 - y1)²]

By applying this formula to the sides of the triangle, we can compare their lengths and ascertain if any two sides are equal.

Calculating Side Lengths

  • Length of AB: Using points A(1,1) and B(2,3):
d_AB = √[(2 - 1)² + (3 - 1)²] = √[1² + 2²] = √5
  • Length of AC: Using points A(1,1) and C(-1,2):
d_AC = √[(-1 - 1)² + (2 - 1)²] = √[(-2)² + 1²] = √5
  • Length of BC: Using points B(2,3) and C(-1,2):
d_BC = √[(-1 - 2)² + (2 - 3)²] = √[(-3)² + (-1)²] = √10

Comparing the lengths, we observe that the length of AB (√5) is equal to the length of AC (√5). This definitively demonstrates that triangle ABC has two sides of equal length. Therefore, triangle ABC is an isosceles triangle.

Conclusion

By meticulously calculating the slopes of the sides and applying the distance formula, we have successfully proven that the points (1,1), (2,3), and (-1,2) form a right-angled isosceles triangle. We first established the presence of a right angle by demonstrating that the product of the slopes of sides AB and AC is -1, confirming their perpendicularity. Subsequently, we proved the isosceles nature of the triangle by showing that sides AB and AC have equal lengths. This analytical approach, grounded in coordinate geometry principles, provides a rigorous and compelling demonstration of the triangle's properties. This exercise not only reinforces the application of slope and distance formulas but also underscores the power of coordinate geometry in solving geometric problems. The ability to translate geometric properties into algebraic relationships and vice versa is a cornerstone of mathematical reasoning, and this proof exemplifies that principle in action.

In summary, the combination of a right angle and two equal sides firmly classifies the triangle formed by the points (1,1), (2,3), and (-1,2) as a right-angled isosceles triangle, a conclusion we have reached through the elegant application of slopes and distance calculations.