Introduction

Identifying whether a triangle is a right-angled triangle based on its vertices is a fundamental concept in both geometry and computer graphics. This article explores various methods to determine if a triangle is right-angled, focusing on vector operations and the Pythagorean theorem.

Using Vectors to Prove a Right Triangle

Consider a triangle with vertices at A(-4, -4), B(1, -9), and C(4, 4). We start by defining the vectors:

vec{AB} (Bx - Ax, By - Ay) (1 - (-4), -9 - (-4)) (5, -5) vec{AC} (Cx - Ax, Cy - Ay) (4 - (-4), 4 - (-4)) (8, 8) vec{BC} (Cx - Bx, Cy - By) (4 - 1, 4 - (-9)) (3, 13)

The dot product of vec{AB} and vec{AC} is:

vec{AB} ? vec{AC}  (5, -5) ? (8, 8)  [5 * 8]   [-5 * 8]  40 - 40  0

Since the dot product is zero, vec{AB} is perpendicular to vec{AC}, confirming that triangle ABC is a right-angled triangle with the right angle at A.

Gradient-Based Approach

The gradient (or slope) of a line segment between two points can be used to determine if the triangle is right-angled. For A(-4, -4), B(1, -9), and C(4, 4): The gradient between A and B is:

Slope of AB (-9 - (-4)) / (1 - (-4)) -5 / 5 -1

The gradient between A and C is:

Slope of AC (4 - (-4)) / (4 - (-4)) 8 / 8 1

Since the product of the gradients of AB and AC is -1, they are perpendicular, and thus, triangle ABC is right-angled at A.

Applying the Pythagorean Theorem

The Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides, can also be used. To check if A(-4, -4), B(1, -9), and C(4, 4) form a right-angled triangle, we calculate the distances between each pair of points:

AB^2 (1 - (-4))^2 (-9 - (-4))^2 5^2 (-5)^2 25 25 50 AC^2 (4 - (-4))^2 (4 - (-4))^2 8^2 8^2 64 64 128 BC^2 (4 - 1)^2 (4 - (-9))^2 3^2 13^2 9 169 178

Since AC^2 AB^2 128 50 178 BC^2, the triangle ABC satisfies the Pythagorean theorem and is thus a right-angled triangle.

General Method to Test Any Triangle

To verify if a given triangle is right-angled, follow these steps:

Calculate the vectors between each pair of points. Compute the dot product of the vectors between each pair of points that share a vertex. If any dot product is zero, the triangle is right-angled at that vertex. Alternatively, compute the gradients of the sides and check if any product is -1, indicating perpendicularity. Or, use the distance formula to calculate each side, then apply the Pythagorean theorem to check for the right angle.

For the vertices A, B, and C, all methods confirm that the triangle is right-angled.

Conclusion

By employing vector operations, gradients, and the Pythagorean theorem, one can effectively determine if a given triangle is right-angled. The methods highlighted in this article provide a robust approach to solving such geometric problems.