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Chapter 6: Problem 12

Find the centroid of the region. The triangle with vertices \((0,0),(1,1)\), and \((2,0)\).

Short Answer

Expert verified
The centroid of the triangle is at \((1, \frac{1}{3})\).

Step by step solution

01

Understand the Centroid Formula for a Triangle

The centroid (geometric center) of a triangle can be found using the formula: \[ C = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \]where \((x_1, y_1), (x_2, y_2), (x_3, y_3)\) are the vertices of the triangle.
02

Identify Triangle Vertices

List the coordinates of the triangle vertices given in the problem: 1. \((0, 0)\)2. \((1, 1)\)3. \((2, 0)\)
03

Apply the Centroid Formula

Substitute the vertices into the centroid formula:\[ C = \left( \frac{0 + 1 + 2}{3}, \frac{0 + 1 + 0}{3} \right) \]This simplifies to:\[ C = \left( \frac{3}{3}, \frac{1}{3} \right) \].
04

Simplify the Coordinates

Calculate the simplified coordinates of the centroid:\[ C = (1, \frac{1}{3}) \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Geometric Center
The geometric center, commonly known as the centroid, is a crucial concept in geometry. It refers to the point that is the average position of all the points in a shape. For a triangle, the centroid is found at the intersection of its medians. A median is a line segment that connects a vertex to the midpoint of the opposite side.
  • The centroid divides each median into two segments, providing a perfect 2:1 ratio with the longer part being between the vertex and the centroid.
  • This special point always lies inside the triangle, regardless of its shape or size.
Understanding the geometric center gives insight into how mass or area is distributed within a shape, and it can be especially useful in contexts like engineering and architecture, where balance is vital.
Triangle Vertices
Triangle vertices are the three corners or points where the sides of a triangle meet. Each triangle has three vertices, and these points are crucial because they define the shape and size of the triangle. In our example, the vertices are given as
  • (0, 0), (1, 1), and (2, 0).
Each vertex is represented by a coordinate in the form
  • (x, y)
which corresponds to its position on the plane.
  • These coordinates are used to calculate other properties of the triangle, such as its centroid.
  • In geometry, understanding the location and role of vertices helps in comprehending the structure of the triangle entirely.
Knowing the vertices allows us to apply formulas and determine various points and distances within the triangle, which are vital for numerous mathematical calculations.
Centroid Formula
The centroid formula provides a simple method to compute the centroid鈥攐r geometric center鈥攐f a triangle using its vertices. The formula is vital in solving different geometric problems, particularly in understanding how a triangle's area is balanced.Let's break it down. For any triangle with vertices
  • (x_1, y_1), (x_2, y_2), (x_3, y_3),
the centroid
  • C
can be found using the formula:\[C = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right)\]This formula works by averaging the x-coordinates and the y-coordinates of the triangle's vertices. As a result, we get a new set of coordinates鈥攖hose of the centroid:
  • In our previous example, substituting the given coordinates gives us the centroid at (1, \( \frac{1}{3} \)).
  • This point is where the three medians intersect, a testament to the balance of areas in the triangle.

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Most popular questions from this chapter

Find the arc length of the catenary \(y=a \cosh (x / a)\) between \(x=0\) and \(x=x_{1}\left(x_{1}>0\right)\).

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