Proving Right-Angled Isosceles Triangle Using Slopes A Comprehensive Guide

Jul 12, 2025 by ADMIN 75 views

Proving a Right-Angled Isosceles Triangle Using Slopes

In geometry, a triangle is one of the most fundamental shapes. Among the various types of triangles, the right-angled isosceles triangle holds a special place due to its unique properties. A right-angled isosceles triangle combines the characteristics of both a right-angled triangle (having one angle of 90 degrees) and an isosceles triangle (having two sides of equal length). To rigorously prove that a given set of points forms such a triangle, we can leverage the concept of slopes. Slopes provide a powerful tool for analyzing the relationships between lines and angles, making them invaluable in geometric proofs. In this article, we will delve into the step-by-step process of using slopes to demonstrate that the points (1,1), (2,3), and (-1,2) indeed form a right-angled isosceles triangle. This method not only confirms the nature of the triangle but also enhances our understanding of coordinate geometry principles. By the end of this discussion, you will be equipped with the knowledge to apply similar techniques to other geometric problems, reinforcing your grasp of mathematical reasoning and proof construction.

Understanding Slopes and Their Significance

To effectively use slopes in proving geometric properties, it is crucial to first understand the concept of slope and its significance in coordinate geometry. The slope of a line is a measure of its steepness and direction on a two-dimensional plane. Mathematically, the slope (often denoted as m) is defined as the change in the y-coordinate divided by the change in the x-coordinate between any two points on the line. The formula to calculate the slope (m) between two points (x₁, y₁) and (x₂, y₂) is given by:

m = (y₂ - y₁) / (x₂ - x₁)

The slope provides valuable information about the line's orientation. A positive slope indicates that the line is increasing (going upwards) as we move from left to right, while a negative slope indicates that the line is decreasing (going downwards). A slope of zero signifies a horizontal line, and an undefined slope (division by zero) represents a vertical line. Furthermore, the relationship between slopes can reveal the angle between two lines. Two lines are parallel if and only if they have the same slope. Two lines are perpendicular if and only if the product of their slopes is -1. This property is particularly important when dealing with right-angled triangles, as it allows us to verify the presence of a 90-degree angle. In the context of our problem, understanding these slope properties is essential for proving that the triangle formed by the given points is indeed a right-angled triangle. By calculating the slopes of the sides and examining their relationships, we can rigorously determine whether the triangle possesses the defining characteristic of a right angle.

Step 1: Calculate the Slopes of the Sides

In order to prove that the points (1,1), (2,3), and (-1,2) form a right-angled isosceles triangle, the first crucial step is to calculate the slopes of the lines connecting these points. Let's denote the points as A(1,1), B(2,3), and C(-1,2). We need to find the slopes of the line segments AB, BC, and CA. The slope of a line segment between two points (x₁, y₁) and (x₂, y₂) is given by the formula:

m = (y₂ - y₁) / (x₂ - x₁)

Slope of AB

Using the coordinates of points A(1,1) and B(2,3), we can calculate the slope of line segment AB:

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

Slope of BC

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

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

Slope of CA

Finally, we calculate the slope of line segment CA using the coordinates of points C(-1,2) and A(1,1):

m_CA = (1 - 2) / (1 - (-1)) = -1 / 2 = -1/2

By performing these calculations, we have determined the slopes of the three sides of the triangle. These slopes will be instrumental in the subsequent steps, where we will use them to verify the right-angled and isosceles properties of the triangle. The next step involves checking for perpendicularity by examining the product of the slopes. If the product of the slopes of any two sides is -1, then those sides are perpendicular, indicating a right angle. This is a critical check in confirming that the triangle is indeed a right-angled triangle. The precise calculation of these slopes sets the foundation for the geometric proof that follows.

Step 2: Check for Perpendicularity

After calculating the slopes of the sides of the triangle formed by the points A(1,1), B(2,3), and C(-1,2), the next crucial step is to check for perpendicularity. This step is essential in determining whether the triangle is a right-angled triangle. Two lines are perpendicular if and only if the product of their slopes is -1. We have already calculated the slopes of the sides AB, BC, and CA as follows:

m_AB = 2
m_BC = 1/3
m_CA = -1/2

Now, we will examine the products of the slopes of each pair of sides to see if any pair satisfies the perpendicularity condition.

Checking AB and BC

Let's multiply the slopes of AB and BC:

m_AB * m_BC = 2 * (1/3) = 2/3

Since 2/3 is not equal to -1, sides AB and BC are not perpendicular.

Checking BC and CA

Next, we multiply the slopes of BC and CA:

m_BC * m_CA = (1/3) * (-1/2) = -1/6

Since -1/6 is not equal to -1, sides BC and CA are not perpendicular.

Checking CA and AB

Finally, we multiply the slopes of CA and AB:

m_CA * m_AB = (-1/2) * 2 = -1

Here, the product of the slopes of CA and AB is -1, which confirms that sides CA and AB are perpendicular. This means that the angle between sides CA and AB is a right angle (90 degrees). Therefore, triangle ABC is a right-angled triangle. The successful identification of a right angle is a significant milestone in our proof, as it validates one of the key characteristics of a right-angled triangle. The next step will involve determining whether the triangle is also isosceles, which means it has two sides of equal length. To verify this, we will calculate the lengths of the sides and compare them.

Step 3: Calculate the Lengths of the Sides

Having established that the triangle formed by the points A(1,1), B(2,3), and C(-1,2) is a right-angled triangle, the next step is to determine whether it is also an isosceles triangle. An isosceles triangle has two sides of equal length. To verify this property, we need to calculate the lengths of the sides AB, BC, and CA using the distance formula. The distance d between two points (x₁, y₁) and (x₂, y₂) in a coordinate plane is given by:

d = √((x₂ - x₁)² + (y₂ - y₁)²)

Length of AB

Using the coordinates of points A(1,1) and B(2,3), we calculate the length of side AB:

AB = √((2 - 1)² + (3 - 1)²) = √(1² + 2²) = √(1 + 4) = √5

Length of BC

Next, we calculate the length of side BC using the coordinates of points B(2,3) and C(-1,2):

BC = √((-1 - 2)² + (2 - 3)²) = √((-3)² + (-1)²) = √(9 + 1) = √10

Length of CA

Finally, we calculate the length of side CA using the coordinates of points C(-1,2) and A(1,1):

CA = √((1 - (-1))² + (1 - 2)²) = √((2)² + (-1)²) = √(4 + 1) = √5

By performing these calculations, we have found the lengths of the three sides of the triangle. We observe that AB = √5 and CA = √5, which means that the lengths of sides AB and CA are equal. This confirms that triangle ABC is an isosceles triangle. Now that we have established both the right-angled and isosceles properties, we can confidently conclude that the triangle formed by the given points is indeed a right-angled isosceles triangle. The combination of these two properties makes this triangle a special case with unique characteristics. The final step is to summarize our findings and state the conclusion based on the calculations and analysis performed.

Step 4: Conclude that the Triangle is Right-Angled Isosceles

Having meticulously calculated the slopes and lengths of the sides of the triangle formed by the points A(1,1), B(2,3), and C(-1,2), we are now in a position to draw a firm conclusion. In the previous steps, we have systematically demonstrated the key properties that define a right-angled isosceles triangle. Let's recap the critical findings:

Slopes: We calculated the slopes of the sides AB, BC, and CA and found that m_AB = 2, m_BC = 1/3, and m_CA = -1/2.
Perpendicularity: By examining the products of the slopes, we determined that m_CA * m_AB = -1, indicating that sides CA and AB are perpendicular. This confirms that angle CAB is a right angle (90 degrees), establishing that the triangle is a right-angled triangle.
Side Lengths: We used the distance formula to calculate the lengths of the sides and found that AB = √5, BC = √10, and CA = √5. The equality of AB and CA (both √5) demonstrates that the triangle is an isosceles triangle, having two sides of equal length.

Combining these findings, we have conclusively proven that triangle ABC possesses both the characteristics of a right-angled triangle and an isosceles triangle. Therefore, we can confidently state that the points (1,1), (2,3), and (-1,2) form a right-angled isosceles triangle. This conclusion is supported by rigorous mathematical calculations and the application of fundamental concepts in coordinate geometry. The use of slopes and the distance formula provided a robust method for analyzing the geometric properties of the triangle and arriving at a definitive result. This exercise not only validates the specific case but also reinforces the broader application of these techniques in solving geometric problems and understanding spatial relationships.

Conclusion

In conclusion, using the principles of coordinate geometry, we have successfully proven that the points (1,1), (2,3), and (-1,2) form a right-angled isosceles triangle. By calculating the slopes of the sides and verifying that the product of the slopes of sides CA and AB is -1, we confirmed the presence of a right angle. Additionally, by calculating the lengths of the sides using the distance formula, we demonstrated that sides AB and CA are of equal length, thus confirming the isosceles property. The combination of these two key characteristics—a right angle and two equal sides—unambiguously establishes that the triangle is indeed a right-angled isosceles triangle. This proof illustrates the power of using slopes and distances in coordinate geometry to analyze and verify geometric properties. The step-by-step approach, from calculating slopes to comparing side lengths, provides a clear and logical framework for solving similar geometric problems. The understanding of these principles not only enhances mathematical proficiency but also fosters a deeper appreciation for the elegance and precision of geometric proofs. This method can be applied to a variety of other geometric problems, reinforcing the fundamental concepts of coordinate geometry and spatial reasoning.