Question

Find the area of the triangle formed by the line \(3 \mathrm{x}-4 \mathrm{y}+12=0\) with the coordinate axes. (1) 6 units \(^{2}\) (2) 12 units \(^{2}\) (3) 1 units \(^{2}\) (4) 36 units \(^{2}\)

Short answer

Answer: 6 square units.

Step-by-step solution

Find the points of intersection with the axes

To find the intersection points of the line with the x and y axes, we will need to find the x and y values when the other is equal to 0. For x-axis (y = 0): \(3x - 4(0) + 12 = 0 \Rightarrow 3x = -12 \Rightarrow x = -4\) So the intersection point with the x-axis is (-4, 0). For y-axis (x = 0): \(3(0) - 4y + 12 = 0 \Rightarrow -4y = -12 \Rightarrow y = 3\) So the intersection point with the y-axis is (0, 3).

Calculate the area of the triangle

Now that we have the points of intersection, we can calculate the area of the triangle formed by the points A (-4, 0), B (0, 3), and C (0, 0). The area of a triangle can be found using the formula: Area = \(\frac{1}{2} \cdot \text{base} \cdot \text{height}\) In this case, the base of the triangle is the distance along the x-axis between points A and C, and the height is the distance along the y-axis between points B and C. We will calculate the area using these distances: Base = \(\lvert -4 - 0 \rvert = 4\) units Height = \(\lvert 3 - 0 \rvert = 3\) units Area = \(\frac{1}{2} \cdot 4 \cdot 3 = \frac{1}{2} \cdot 12 =\underline{6}\) square units Therefore, the area of the triangle formed by the line \(3x-4y+12=0\) with the coordinate axes is 6 square units (option 1).

Key concepts

Coordinate Axes

The coordinate axes are the foundation of coordinate geometry, dividing the plane into four quadrants and serving as a reference for locating points in a Cartesian plane. Specifically, the two perpendicular lines are the X-axis (horizontal) and Y-axis (vertical). Every point in this system is defined by an ordered pair of numbers, known as coordinates, which represent the distances from the respective axes.

When a line equation, like in our exercise, is set equal to zero, it commonly represents the intersection of the line with one of the coordinate axes, either X or Y, depending on which variable is present. The X-axis intersections occur when the Y-value is zero, and vice versa for the Y-axis. Understanding how a line intersects with the axes allows you to identify significant points that aid in further geometric calculations, such as determining the vertices of a triangle.

Intersection Points

Intersection points are where a line or curve meets another line, curve, or plot such as the coordinate axes. Finding these points is crucial when solving many geometrical problems, including the one in our exercise.

To find the point where a line intersects the X-axis, we set the Y-coordinate to zero and solve for X. Similarly, when finding where a line crosses the Y-axis, we set X to zero. These intersection points are vital as they often represent corners or vertices of polygons and are used to determine the shapes' properties.

In coordinate geometry, these intersection points provide a way to concretely describe the position of shapes, which can then allow for the calculation of various attributes, like distance, midpoint, slope, and in our case, areas of geometric figures.

Triangle Area Calculation

Calculating the area of a triangle can be approached in various ways, depending on the information available. One standard method is by using the formula \[ \text{Area} = \frac{1}{2} \cdot \text{base} \cdot \text{height} \], which calls for the length of the base and the height of the triangle.

In our exercise, the triangle is formed by the intersection of a line with the coordinate axes, creating a right-angled triangle. This simplifies the situation because one side lies on the X-axis and one side on the Y-axis, making the length of these sides easily measurable without requiring more complex geometry or trigonometry.

For non-right-angled triangles, alternative methods include Heron's formula or the shoelace formula, applicable when dealing with vertices coordinates. However, for triangles on a Cartesian plane with one vertex at the origin, and others along the axes, the intersection points can directly give the lengths of sides that serve as the base and height. This makes the calculation straightforward and ensures a quick method to obtain the area.