Question

A triangle is formed by the lines \(\mathrm{x}+\mathrm{y}=8, \mathrm{X}\) -axis and Y-axis. Find its centroid. (1) \(\left(\frac{8}{3}, \frac{8}{3}\right)\) (2) \((8,8)\) (3) \((4,4)\) (4) \((0,0)\)

Short answer

Answer: The coordinates of the centroid of the triangle are (8/3, 8/3).

Step-by-step solution

Identify the vertices of the triangle.

The triangle is formed by the equations x + y = 8, x = 0, and y = 0. We will find the coordinates of the vertices formed by the intersection of these lines: Intersection of x + y = 8 and x = 0: x = 0 y = 8 Vertex A: (0, 8) Intersection of x + y = 8 and y = 0: y = 0 x = 8 Vertex B: (8, 0) Intersection of x = 0 and y = 0: x = 0 y = 0 Vertex C: (0, 0) The triangle's vertices are A(0, 8), B(8, 0), and C(0, 0).

Calculate the coordinates of the centroid.

The centroid G can be determined from the coordinates of the vertices as follows: Gx = (x1 + x2 + x3)/3 Gy = (y1 + y2 + y3)/3 Using the coordinates of the vertices A(0, 8), B(8, 0), and C(0, 0): Gx = (0 + 8 + 0) / 3 = 8 / 3 Gy = (8 + 0 + 0) / 3 = 8 / 3 G = (8/3, 8/3), which is option (1). So, the centroid of the triangle is (8/3, 8/3).

Key concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, plays a crucial role in modern mathematics. It allows the precise plotting of points, lines, and shapes on a two-dimensional plane using a set of coordinates. These coordinates typically involve an x and a y value that establish position in relation to two intersecting lines called axes, usually labeled the X-axis and Y-axis.

In the context of finding the centroid of a triangle, coordinate geometry simplifies the process by providing a way to calculate the exact point that represents the average position of all the points in the triangle. The fundamental formula for finding the centroid involves averaging the x and y coordinates of the triangle's vertices. Understanding how to apply coordinate geometry concepts to such problems is essential for students studying geometry, physics, engineering, and many other fields.

Vertex of a Triangle

The vertex of a triangle is a point where two sides of the triangle meet. A triangle has three vertices, commonly denoted as A, B, and C. These vertices serve as fundamental reference points for many geometrical calculations and properties concerning triangles. For instance, in the context of coordinate geometry, we assign coordinates to each vertex to facilitate mathematical analysis of the triangle's properties.

When working with problems involving vertices, it is important to determine their coordinates accurately, as they form the basis for calculating the centroid, area, and other important attributes of the triangle. Recognizing how to identify and work with the vertices in various contexts is key to mastering the basics of triangle geometry.

Intersection of Lines

When two lines cross each other on a plane, they intersect at a point, which has a set of coordinates that can be calculated based on the equations of the lines. In many geometric problems, finding the intersection of lines is a fundamental step, often leading to the identification of important points such as the vertices of a triangle.

The process of determining the intersection can involve solving a system of equations that represents the lines in question. For example, if two lines are defined by the equations y = mx + c and y = nx + d, setting them equal to each other, since their y-coordinates at the point of intersection are the same, gives a solution for x, which can then be substituted back into one of the equations to find the corresponding y-coordinate. Understanding how to find the intersection of lines is an essential skill in coordinate geometry and is widely applicable in various fields, including computer graphics, navigation, and urban planning.