Triangle ABC Right Angle Determination A Comprehensive Analysis

Jul 13, 2025 by ADMIN 64 views

Is triangle $ABC$ with vertices at $A(-2,3)$ , $B(-3,-6)$ , and $C(2,-1)$ a right triangle? If so, which angle is the right angle? This is a classic geometry problem that can be solved using the concepts of slope and the Pythagorean theorem. In this article, we will delve into a step-by-step solution, providing a clear and concise explanation to help you understand the underlying principles. Understanding how to determine if a triangle is a right triangle is fundamental in geometry and has practical applications in various fields, including engineering, architecture, and computer graphics. Let's embark on this geometrical journey together!

Understanding Right Triangles

Before we dive into the specifics of triangle $ABC$ , let's establish a solid understanding of what constitutes a right triangle. A right triangle is a triangle that has one angle measuring exactly 90 degrees. This angle is often referred to as a right angle. The side opposite the right angle is called the hypotenuse, and it is the longest side of the triangle. The other two sides are called legs. Right triangles hold a special place in geometry due to their unique properties and the numerous theorems associated with them, most notably the Pythagorean theorem.

The Pythagorean theorem is a cornerstone of geometry and is exclusively applicable to right triangles. It states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (the legs). Mathematically, this can be expressed as: $a^2 + b^2 = c^2$ , where $c$ represents the length of the hypotenuse, and $a$ and $b$ represent the lengths of the legs. This theorem is a powerful tool for determining if a triangle is a right triangle, as well as for calculating the lengths of sides in a right triangle if other side lengths are known. Furthermore, the concept of perpendicularity is crucial in identifying right angles. Two lines are perpendicular if they intersect at a right angle. This property translates into slopes: two lines are perpendicular if and only if the product of their slopes is -1. This relationship between slopes and perpendicularity provides an alternative method for verifying the presence of a right angle in a triangle.

Understanding these fundamental concepts—the definition of a right triangle, the Pythagorean theorem, and the relationship between perpendicular lines and slopes—is essential for tackling the problem at hand. We will leverage these principles to analyze triangle $ABC$ and definitively determine whether it qualifies as a right triangle and, if so, identify the right angle. Mastering these concepts opens doors to solving a wide range of geometric problems.

Calculating Slopes of the Sides

To determine if triangle $ABC$ is a right triangle, we can utilize the concept of perpendicular lines. As mentioned earlier, two lines are perpendicular if the product of their slopes is -1. Therefore, we need to calculate the slopes of the sides of the triangle and check if any two sides are perpendicular. The slope of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula:

$m = (y_2 - y_1) / (x_2 - x_1)$

Let's calculate the slopes of the sides $AB$ , $BC$ , and $CA$ using the given coordinates $A(-2,3)$ , $B(-3,-6)$ , and $C(2,-1)$ .

Slope of AB ( $m_{AB}$ ):
- Using points $A(-2,3)$ and $B(-3,-6)$ :
- $m_{AB} = (-6 - 3) / (-3 - (-2)) = (-9) / (-1) = 9$

The slope of side $AB$ is 9. This positive slope indicates that the line segment rises as it moves from left to right. The steepness of the line is represented by the magnitude of the slope, with a higher absolute value indicating a steeper slope. In this case, a slope of 9 signifies a relatively steep incline for side $AB$ .

Slope of BC ( $m_{BC}$ ):
- Using points $B(-3,-6)$ and $C(2,-1)$ :
- $m_{BC} = (-1 - (-6)) / (2 - (-3)) = (5) / (5) = 1$

The slope of side $BC$ is 1. A slope of 1 means that for every unit increase in the x-coordinate, the y-coordinate also increases by one unit. This represents a moderate incline, less steep than side $AB$ . The positive slope again indicates an upward trend from left to right, but the gentler slope suggests a less drastic change in height compared to side $AB$ .

Slope of CA ( $m_{CA}$ ):
- Using points $C(2,-1)$ and $A(-2,3)$ :
- $m_{CA} = (3 - (-1)) / (-2 - 2) = (4) / (-4) = -1$

The slope of side $CA$ is -1. A negative slope indicates that the line segment descends as it moves from left to right. The slope of -1 signifies a moderate decline, mirroring the steepness of side $BC$ but in the opposite direction. This means that for every unit increase in the x-coordinate, the y-coordinate decreases by one unit.

Now that we have the slopes of all three sides, we can proceed to check for perpendicularity by examining the products of the slopes. This is the crucial next step in determining if triangle $ABC$ is a right triangle.

Checking for Perpendicularity

Now that we have calculated the slopes of the sides of triangle $ABC$ , we can determine if any two sides are perpendicular. Recall that two lines are perpendicular if and only if the product of their slopes is -1. We will examine the products of the slopes of each pair of sides:

Product of slopes of AB and BC:
- $m_{AB} * m_{BC} = 9 * 1 = 9$

The product of the slopes of sides $AB$ and $BC$ is 9. Since this product is not equal to -1, sides $AB$ and $BC$ are not perpendicular. This means that angle $B$ is not a right angle. For two lines to be perpendicular, their slopes must be negative reciprocals of each other, and the product of their slopes must equal -1. In this case, the product is a positive number significantly greater than -1, indicating a substantial deviation from perpendicularity.

Product of slopes of BC and CA:
- $m_{BC} * m_{CA} = 1 * (-1) = -1$

The product of the slopes of sides $BC$ and $CA$ is -1. This confirms that sides $BC$ and $CA$ are perpendicular. Therefore, the angle between these two sides, which is angle $C$ , is a right angle. This finding is significant as it directly answers the second part of our question: if the triangle is a right triangle, which angle is the right angle? The result unequivocally points to angle $C$ .

Product of slopes of CA and AB:
- $m_{CA} * m_{AB} = (-1) * 9 = -9$

The product of the slopes of sides $CA$ and $AB$ is -9. This product is also not equal to -1, indicating that sides $CA$ and $AB$ are not perpendicular. Consequently, angle $A$ is not a right angle. Similar to the case with sides $AB$ and $BC$ , the product's deviation from -1 demonstrates a clear lack of perpendicularity between sides $CA$ and $AB$ .

Based on these calculations, we have definitively determined that sides $BC$ and $CA$ are perpendicular, making angle $C$ the right angle in triangle $ABC$ . This conclusion allows us to answer the original question with certainty.

Conclusion: Triangle ABC is a Right Triangle

After meticulously calculating the slopes of the sides of triangle $ABC$ and checking for perpendicularity, we have arrived at a conclusive answer. The product of the slopes of sides $BC$ and $CA$ is -1, which confirms that these two sides are indeed perpendicular. This means that the angle formed at the intersection of sides $BC$ and $CA$ , which is angle $C$ , is a right angle.

Therefore, triangle $ABC$ is a right triangle, and the right angle is angle $C$ .

This problem demonstrates the power of using slopes to analyze geometric figures. By understanding the relationship between slopes and perpendicularity, we can efficiently determine if a triangle is a right triangle without having to rely solely on the Pythagorean theorem. However, it's worth noting that the Pythagorean theorem could also be used to verify our result. We could calculate the lengths of the sides using the distance formula and then check if the square of the longest side is equal to the sum of the squares of the other two sides. If the Pythagorean theorem holds true, it would further confirm that triangle $ABC$ is a right triangle.

In summary, we have successfully determined that triangle $ABC$ with vertices $A(-2,3)$ , $B(-3,-6)$ , and $C(2,-1)$ is a right triangle, and the right angle is angle $C$ . This comprehensive analysis provides a clear understanding of the concepts and methods used to solve this type of problem.