Problem 58 - Algebra and Trigonometry

Question

Find the area of the triangle formed by the coordinate axes and the line $$2 y+3 x-6=0$$

Step-by-Step Solution

Answer

The area is 3 square units.

Step 1: Find the Intercepts

To determine the area of a triangle formed by a line with the coordinate axes, we start by finding where the line intersects the x-axis and y-axis. This line is given by the equation \(2y + 3x - 6 = 0\). To find the x-intercept, set \(y = 0\) and solve \(3x - 6 = 0\). To find the y-intercept, set \(x = 0\) and solve \(2y - 6 = 0\).

Step 2: Solving for x-Intercept

Set \(y = 0\) in the equation \(2y + 3x - 6 = 0\), resulting in \(3x - 6 = 0\). Solving for \(x\), we get \(3x = 6\), so \(x = 2\). Therefore, the line intersects the x-axis at \((2, 0)\).

Step 3: Solving for y-Intercept

Set \(x = 0\) in the equation \(2y + 3x - 6 = 0\), resulting in \(2y - 6 = 0\). Solving for \(y\), we get \(2y = 6\), so \(y = 3\). Therefore, the line intersects the y-axis at \((0, 3)\).

Step 4: Calculate the Area of the Triangle

The triangle is formed with vertices \((0, 0)\), \((2, 0)\), and \((0, 3)\). The base of the triangle along the x-axis is 2 units, and the height along the y-axis is 3 units. The area \(A\) of the triangle is given by the formula \(A = \frac{1}{2} \times \text{base} \times \text{height}\). Therefore, the area is \([\frac{1}{2} \times 2 \times 3 = 3]\) square units.

Key Concepts

Intercepts

Intercepts are the points where a line crosses the axes on a graph. These points are crucial for solving many coordinate geometry problems, such as determining the area of a triangle formed with the coordinate axes.
Starting with the **x-intercept**, it is the point where the line crosses the x-axis, which means the y-coordinate is zero. To find it, substitute \(y=0\) into the line's equation. For example, with the equation \(2y + 3x - 6 = 0\), setting \(y=0\) gives us \(3x - 6 = 0\). Solving this, \(x = 2\), so our x-intercept is \((2, 0)\).
Next is the **y-intercept**, where the line crosses the y-axis, making the x-coordinate zero. Substitute \(x=0\) into the line's equation. Using the same example, set \(x=0\), resulting in \(2y - 6 = 0\). Solving this, \(y = 3\), so the y-intercept is \((0, 3)\).
Both intercepts are essential for forming geometric shapes such as triangles with the coordinate axes.

Triangle Area

Finding the area of a triangle, especially one formed by a line with coordinate axes, is a straightforward process if we understand its base and height.
The vertices of the triangle in this problem are given by the intercepts and the origin:

• Base: The length along the x-axis between the intercept and the origin.
• Height: The length along the y-axis from the intercept to the origin.

Using the triangle formed by

• the origin \((0, 0)\)
• the x-intercept \((2, 0)\)
• the y-intercept \((0, 3)\)

The **base** of the triangle is 2 (distance between \((0, 0)\) and \((2, 0)\)), and the **height** is 3 (distance between \((0, 0)\) and \((0, 3)\)).
For a right-angled triangle, like in this case, the area \(A\) is given by \[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Performing the calculation: \( A = \frac{1}{2} \times 2 \times 3 = 3 \) square units.

Equation of Line

The equation of a line is fundamental in coordinate geometry. It generally describes all points along a line in the plane.
In an equation like \(2y + 3x - 6 = 0\), each term has an important role:

• The term \(2y\) represents the influence of the y-coordinate.
• The term \(3x\) represents the influence of the x-coordinate.
• The constant \(-6\), also called the intercept, adjusts the line's position on the graph.

Such equations can be manipulated to make certain calculations easier:
If we rearrange the equation to the slope-intercept form \(y = mx + c\), it becomes \[ y = -\frac{3}{2}x + 3 \]
where \(-\frac{3}{2}\) is the slope and 3 is the y-intercept. This form makes it simpler to visualize and understand the line's behavior, guiding us in solving for specific points and relationships in geometric configurations.