91Ó°ÊÓ

When examining the relationships between lines on a coordinate plane, one of the most critical concepts is that of perpendicular lines. These are lines that meet or intersect each other at a right angle (90 degrees). In the context of a right triangle, the sides that form the right angle are represented by two lines that are perpendicular to one another.

To identify whether two lines are perpendicular, one can use their slopes, a fundamental trait that quantifies their steepness. For lines with non-vertical slopes, if the product of the slopes of two lines is -1, those lines are perpendicular. This relationship arises from the fact that the slopes of perpendicular lines are negative reciprocals of each other.

Applying this concept to the triangle with vertices at points A(-1,5), B(3,7), and C(5,3), we compute the slopes of the lines AB and BC, and as demonstrated in the solution, their product equals -1, confirming that AB and BC are indeed perpendicular to each other and form a right angle at vertex B.

The vertices of a triangle are the points where its sides intersect. In coordinate geometry, these points are defined by their coordinates on the Cartesian plane. When dealing with the task of proving whether three points can form a right triangle, an effective approach is to calculate the slopes between each pair of points (vertices).

For our example, the points A(-1,5), B(3,7), and C(5,3) are suspected to form the vertices of the triangle. By calculating the slopes of segments AB, BC, and AC, we can gain insight into the relationships between the sides. If two sides are found to be perpendicular (forming a right angle), then this validates the existence of a right triangle and identifies the right-angle vertex. In our case, the calculations in the step by step solution have confirmed that point B is the right-angle vertex due to the perpendicularity of lines AB and BC.

The slope formula is pivotal in coordinate geometry and is especially useful when dealing with triangles and their orientation. The formula for calculating the slope of a line segment between two points, (x1, y1) and (x2, y2), is (m = (y2 - y1) / (x2 - x1)). This slope is a measure of the 'steepness' or the inclination of the line. A positive slope means the line is ascending from left to right, while a negative slope indicates a descent. A slope of zero implies a horizontal line, and an undefined slope (when (x2 - x1) is zero) corresponds to a vertical line.

By applying this formula to create the steps in the given solution, we established the slopes of the segments AB, BC, and AC. These slope values were then used to determine that lines AB and BC are perpendicular, confirming that the triangle formed by the points A, B, and C is indeed a right triangle. Remembering and utilizing the slope formula is essential for solving a wide array of problems in geometry, making it an invaluable tool for students.