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Coordinate geometry, or analytic geometry, is the study of geometry using a coordinate system. This is used to simplify the process of understanding geometric shapes like points, lines, and curves through algebraic equations. For a point in the coordinate plane, two values are needed—the x-coordinate (abscissa) and the y-coordinate (ordinate). In the context of the exercise, we have points A, B, and C, all of which are represented as coordinates in the first quadrant of the Cartesian plane. Points A and B have known coordinates: A(1,2) and B(3,1). The task involves finding the x-coordinate for the unknown point C, using known geometric properties.
To effectively solve problems in coordinate geometry, one often utilizes algebraic methods to find distances, midpoints, and slopes, all of which further help in understanding the precise location and characteristics of geometric shapes.

Triangles have unique properties that make them a fundamental shape in geometry. In this exercise, the triangle is right-angled at vertex A, meaning it has one 90-degree angle. Such triangles exhibit the Pythagorean Theorem, which relates the lengths of their sides.The vertices A, B, and C represent the corners of the triangle. Due to the right-angle at \(A\), the sides AB and AC are perpendicular, necessitating distinct calculations for lengths and slopes. Working within these triangle properties helps determine lengths of sides using given conditions, like the area or sides' slopes. Understanding a triangle's properties also involves knowing its angles, side ratios, and how they translate into coordinates when the vertices are placed in a coordinate plane. For instance, an equation like \(y = 2x\) for line AC arises from knowing one side's slope (as perpendicular to AB) and its geometric context.

The area of a triangle is the space enclosed within its three vertices. For a right-angled triangle like \(\triangle ABC\), the area can easily be calculated using its base and height. The formula \[ \text{Area} = \frac{1}{2} imes \text{Base} imes \text{Height} \] is applied, where the base is the length of line segment AB and the height is determined by the line perpendicular to it, AC.In this exercise, the given area is \(5\sqrt{5}\) square units, which allows for the calculation of unknown length AC. Knowing this area and applying the geometric formula assists in finding the precise length of AC, thereby helping in determining the x-coordinate of vertex C. Understanding this property aids in not just solving for missing lengths but also reinforces the importance of accurately plotting coordinates in a geometric context.

Slopes are a critical concept in coordinate geometry, providing essential information about line direction and steepness. The slope of a line formed by two points, such as between A and B, is calculated by the formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]For line AB, this yields a slope of \(-\frac{1}{2}\). In the case of a perpendicular line, e.g., line AC, the slope will be the negative reciprocal of AB's slope, thus making it 2.The equation of a line can be determined if the slope and a point on the line are known. This concept is utilized in the exercise to find line AC's equation. Understanding slopes is vital in solving geometric problems like the placement of vertex C: it provides a framework to determine how one line relates to another in terms of direction and placement through the coordinate plane.