Question

Find the area of the triangle formed by the line \(5 \mathrm{x}-3 \mathrm{y}+15=0\) with coordinate axes. (1) \(15 \mathrm{~cm}^{2}\) (2) \(5 \mathrm{~cm}^{2}\) (3) \(8 \mathrm{~cm}^{2}\) (4) \(\frac{15}{2} \mathrm{~cm}^{2}\)

Short answer

Question: Given the equation of a line, 5x - 3y + 15 = 0, find the area of the triangle formed by the line and the coordinate axes. Answer: To find the area of the triangle formed by the line and the coordinate axes, follow these steps: 1. Find the x and y intercepts of the line. 2. Calculate the lengths of the base and height of the triangle using the intercepts. 3. Calculate the area of the triangle using the formula: \(Area = \frac{1}{2}base \times height\). Substitute the values obtained in the steps above to calculate the area of the triangle.

Step-by-step solution

Find the x and y intercepts of the line

To find the x-intercept, set y=0 in the given equation: \(5x - 3(0) + 15 = 0\). Solve for x to get the x-intercept. Similarly, to find the y-intercept, set x=0 in the equation: \(5(0) - 3y + 15 = 0\). Solve for y to get the y-intercept.

Calculate the lengths of the base and height of the triangle

From Step 1, we have the x and y intercepts, which represent the points where the line meets the x and y axes, respectively. The distance of these intercepts from the origin, will give us the lengths of the base and height of the triangle.

Calculate the area of the triangle

Use the formula for the area of a triangle, which is \(Area = \frac{1}{2}base \times height\). Plug in the values obtained in Step 2 and calculate the area. Finally, compare the calculated area with the provided answer choices to find the correct one.

Key concepts

Coordinate Geometry

Coordination Geometry, also referred to as analytic geometry, is a powerful tool in mathematics that involves the study of geometry using a coordinate system. This branch of geometry allows us to solve problems and prove results by combining algebra and geometry together. It involves placing shapes onto the coordinate plane and using their coordinates to solve for various properties like length, area, and slope.

With regard to triangle area calculation in coordinate geometry, once the vertices of the triangle are known, there are several formulas at our disposal for calculating the area. For example, if we know the coordinates of the three vertices, we can use the Shoelace formula or the determinant of a matrix derived from the vertices’ coordinates. However, when the triangle is formed by a line and the coordinate axes, as seen in our original exercise, we can employ a simpler approach using intercepts for a more straightforward solution.

X-Intercept and Y-Intercept

Understanding x-intercepts and y-intercepts is crucial when working within the realm of coordinate geometry. An x-intercept is the point where a line or curve crosses the x-axis of the coordinate plane, and it is found by setting the y-coordinate to zero and solving for x. Likewise, the y-intercept is the point at which the line or curve crosses the y-axis, which is calculated by setting the x-coordinate to zero and solving for y.

In our textbook exercise, identifying these intercepts is the first step to forming our triangle, as they become the vertices that, along with the origin, form a right-angled triangle on the plane. This information can then be used to determine the base and height of the triangle, which are essential in computing its area. By mastering intercepts, you unlock a vital skill that aids in graphing lines, understanding linear equations, and solving geometric problems.

Geometry Problem Solving

Geometry problem solving in mathematics involves applying geometry principles to find solutions to various shapes and figures' properties. These problems often require a logical approach and sometimes the use of algebraic methods. Crucial to problem solving is the ability to visualize geometric shapes and dissect their properties like angles, lengths, and areas. Effective problem-solving strategies might include drawing diagrams, identifying patterns, and breaking down complex shapes into simpler components.

In the context of finding the area of a triangle formed by a line with the coordinate axes, the strategy would involve a step-by-step method. First, identify the intercepts to determine the triangle's vertices, followed by using these values as lengths of the base and height. Then we apply the fundamental formula for the area of a triangle. By carefully executing each step and ensuring accuracy at every stage, one can efficiently navigate through geometry problems from textbooks, standardized tests, or real-life situations.