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Coordinate geometry is a branch of mathematics that explores the relationship between geometry and algebra using a coordinate system. When dealing with triangles on a coordinate plane, we locate each vertex using an ordered pair of numbers, a coordinate. These numbers pinpoint the exact location of a point, such as a vertex of a triangle, on a plane.
To solve problems involving triangles in coordinate geometry, you often plot the points representing the triangle's vertices on the Cartesian plane. This involves identifying the horizontal (x-axis) and vertical (y-axis) positions of each vertex. For example, given points A(-3,0), B(0,-2), and C(2,3), these coordinates are used to plot triangle ABC. Note how each coordinate gives us the x and y value of each point, essential for visual and analytical purposes.

In geometry, an altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side. This segment represents the shortest distance from the vertex to the base of a triangle. In the context of coordinate geometry, finding the altitude involves calculations using coordinates and line equations.
For triangle ABC, the altitude from vertex B to side AC requires determining the perpendicular distance from point B to the line formed by A and C. First, we need to find the equation of line AC, using its slope which can be calculated as \( m = \frac{y2-y1}{x2-x1} \), giving us \( \frac{3}{5} \). With the slope, the line equation AC is determined, enabling us to substitute B's coordinates into a formula to find this perpendicular distance (the altitude).

Calculating the area of a triangle on a coordinate plane requires knowing the base and the height (altitude) of the triangle. The formula is straightforward: the area is half of the product of the base and the height. When the triangle is plotted on a coordinate plane, any side can be designated as the base.
In the case of triangle ABC, after determining the length of AC (the base), and the altitude from B, you can apply the formula: \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \.\) This formula ensures you consider the two-dimensional space the triangle occupies, allowing you to find its specific area by integrating both its dimensions.

The distance formula is a formula used to find the distance between two points on a coordinate plane. It is derived from the Pythagorean theorem and is essential for various calculations in coordinate geometry, including finding the altitude of triangles.
Given two points, \( (x_1, y_1) \) and \( (x_2, y_2) \), the distance, d, between these points is calculated by the formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \].
This formula enables us to determine not only the length of a side of a triangle but is also adapted into the formula to find the perpendicular distance from a point to a line, crucial for determining altitudes in triangles such as the altitude from vertex B to side AC in triangle ABC.