Question

The following are the steps involved in finding the centroid of the triangle with vertices \((5,4),(6,7)\) and \((1,1)\). Arrange them in sequential order from first to last. (a) The required centroid \((4,4)\). (b) The centroid of the triangle with the vertices \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)\) and \(\left(x_{3}, y_{y}\right)\) is \(\left(\frac{x_{1}+x_{2}+x_{3}}{3}, \frac{y_{1}+y_{2}+y_{3}}{3}\right)\). (c) The centroid is \(\left(\frac{5+6+1}{3}, \frac{4+7+1}{3}\right)\). (1) abc (2) cba (3) bac (4) bca

Short answer

Question: Arrange the given steps in the correct order to find the centroid of a triangle with vertices (5,4), (6,7), and (1,1): a. (4,4) b. Centroid formula: \(\left(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3}\right)\) c. \(\left(\frac{5+6+1}{3},\frac{4+7+1}{3}\right)\) Answer: Option (4) - bca

Step-by-step solution

Given formula for centroid

We have the formula given in option (b) for finding the centroid of a triangle with given vertices \((x_1, y_1), (x_2, y_2),\) and \((x_3, y_3)\). Since this is correct, the first step in our solution will be option (b).

Substitute the vertex coordinates into the formula

Now, using the given vertices \((5,4),(6,7)\) and \((1,1)\), substitute the coordinates into the formula for the centroid. It matches with option (c): \(\left(\frac{5+6+1}{3}, \frac{4+7+1}{3}\right)\). So, the second step will be option (c).

Calculate the centroid coordinates

Finally, calculate the coordinates of the centroid by following the expression given in step (c). It matches with the coordinate of the centroid given in option (a): \((4,4)\). So, the final step in our solution will be option (a). Based on the above, the correct order of steps for this exercise is option (4) - bca.

Key concepts

Geometry

Geometry is the branch of mathematics that deals with the properties, measurements, and relationships of points, lines, angles, surfaces, and solids. In the context of the centroid of a triangle, it's a fundamental concept that describes the point where the three medians of the triangle intersect. The median of a triangle is a line segment connecting a vertex to the midpoint of the opposite side.

Understanding the geometry of a triangle and its properties, such as medians and centroids, is crucial for solving many problems in both theoretical and applied mathematics. The centroid is also known as the geometric center of a triangle, implying it is the balance point at which the triangle can be perfectly balanced on the tip of a pencil.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves the study of geometry using a coordinate system. This method enables the definition of geometric shapes in a numerical way and simplifies the process of calculating properties like distance, slope, and area.

In coordinate geometry, locating the centroid of a triangle involves finding the average of the x-coordinates and y-coordinates of the triangle's vertices. The formula provided in the problem \(\left(\frac{x_{1}+x_{2}+x_{3}}{3}, \frac{y_{1}+y_{2}+y_{3}}{3}\right)\) represents this process mathematically. Students learning this discipline will find that being familiar with plotting points and calculating averages is essential for deriving the centroid's coordinates in a Cartesian plane.

Math Problem Solving

Math problem solving is an intellectual exercise that involves understanding and solving mathematical problems by applying various strategies and concepts. The key to successful problem solving is to identify the required information, understand the problem's structure, and implement a step-by-step approach to arrive at a solution.

When solving a math problem like finding a triangle's centroid, it's crucial to proceed methodically. Start with understanding the formula, then substitute the specific values from the problem, and finally, compute the results. This systematic approach helps in reducing errors and increases the chance of reaching the correct solution. By organizing the steps in sequential order, as with the options (b), (c), and (a) in the problem provided, students can ensure that their method is both logical and efficient. The further advice to enhance understanding could include: Visualizing the problem by sketching the triangle and its medians, checking the work by using an alternative method if available, and relating the problem to real-world situations where the centroid has practical applications, like in engineering or design.