A line intersects \(\mathrm{X}\) -axis and Y-axis at \(\mathrm{A}\) and \(\mathrm{B}\) respectively. If \(\mathrm{AB}=15\) and \(\underline{\mathrm{AB}}\) makes a triangle of area 54 units with coordinate axes, then the equation of \(\underline{A B}\) is \(\ldots .\) (a) \(4 x \pm 3 y=36\) or \(3 x \pm 4 y=36\) (b) \(4 x \pm 3 y=24\) or \(3 x \pm 4 y=24\) (c) \(-4 \mathrm{x} \pm 3 \mathrm{y}=24\) or \(-3 \mathrm{x} \pm 4 \mathrm{y}=24\) (d) \(-4 x \pm 3 y=12\) or \(-3 x \pm 4 y=12\)

Understanding the equation of a line is fundamental in coordinate geometry. It represents a straight path that extends in both directions without end. To construct the equation of a line, various forms can be utilized, such as the two-point form, the point-slope form, and the slope-intercept form.

The two-point form, used in our textbook solution example, requires knowing the coordinates of two points that the line passes through, typically denoted as \( (x_1, y_1) \) and \( (x_2, y_2) \). The formula is as follows: \[ \frac{y - y_1}{x - x_1} = \frac{y_2 - y_1}{x_2 - x_1} \] This equation enables us to find the unique line that passes through both given points.

When solving such problems, it's crucial to simplify and rearrange the equation to a recognizable form, such as the standard form \( Ax + By = C \) or the slope-intercept form \( y = mx + b \) where \( m \) is the slope and \( b \) is the y-intercept. By understanding how to derive and manipulate these equations, students can easily determine any line's direction, location, and the relationships between lines, such as parallelism or perpendicularity.

The distance formula is vital in coordinate geometry and is a direct application of the Pythagorean Theorem to find the length between two points on a coordinate plane. For any two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) on the Cartesian plane, the distance between them, denoted as \( AB \) is given by the equation: \[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

This formula is derived from the concept of creating a right-angled triangle with the points and measuring the hypotenuse, which represents the distance between these points. In the textbook solution, this distance was an essential piece of information used to establish the relationship between the coordinates of points A and B when combined with the information about the area of the triangle. The better students understand the distance formula, the more accurately they can measure distances and solve complex problems involving the lengths of segments in geometry.

Determining the area of a triangle is a common task in geometry and can involve various methods depending on the given information. The most straightforward formula is for right-angled triangles where the base and height are perpendicular: \[ Area = \frac{1}{2} \cdot Base \cdot Height \]

In the context of coordinate geometry and our textbook example, when a triangle is formed by the coordinate axes, its sides lie along the X and Y axes, making them perpendicular by default. This property simplifies the area calculation since we can use the coordinates to define the length of the base and the height. In the given problem, combining the knowledge of the length of side AB and the area of triangle ABC, where C is the origin, enables us to establish a system of equations that leads us to the triangle's specific dimensions. With these dimensions, it becomes possible to derive further properties of the triangle, such as the location of its vertices or, as seen in our solution steps, the equation of line AB. Recognizing the implicit relationships between area, distance, and line equations greatly enhances a student's ability to solve geometry problems.