Problem 22 - Thomas Calculus

Question

Finding a centroid Find the centroid of the triangular region cut from the first quadrant by the line \(x+y=3\)

Step-by-Step Solution

Answer

The centroid of the triangle is at (1, 1).

Step 1: Understand the Problem

The problem requires us to find the centroid of a triangular region in the first quadrant, bounded by the line \(x+y=3\). This means looking for the point where the balancing point of this triangular region would be located.

Step 2: Determine the Vertices of the Triangle

The line \(x+y=3\) intersects the x-axis when \(y=0\) at \((3,0)\) and the y-axis when \(x=0\) at \((0,3)\). Therefore, the vertices of the triangle are \((0,0), (3,0), (0,3)\).

Step 3: Use the Centroid Formula for a Triangle

The formula for the centroid \((x_c, y_c)\) of a triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) is:\[x_c = \frac{x_1 + x_2 + x_3}{3}, \quad y_c = \frac{y_1 + y_2 + y_3}{3}\]

Step 4: Calculate the x-coordinate of the Centroid

Substitute the x-coordinates of the vertices into the formula for the centroid's x-coordinate:\[x_c = \frac{0 + 3 + 0}{3} = \frac{3}{3} = 1\]

Step 5: Calculate the y-coordinate of the Centroid

Substitute the y-coordinates of the vertices into the formula for the centroid's y-coordinate:\[y_c = \frac{0 + 0 + 3}{3} = \frac{3}{3} = 1\]

Step 6: State the Centroid Coordinates

The centroid \((x_c, y_c)\) of the triangle is at \((1, 1)\).

Key Concepts

Geometry

Geometry is a branch of mathematics that deals with shapes, sizes, and the properties of space. One fundamental concept in geometry is the idea that shapes like triangles, squares, and circles have specific properties that can be easily calculated.
For a triangle, the centroid is a crucial point of interest. The centroid is considered the triangle's center of mass or balance point.

• It is the intersection of its medians, which are the line segments that connect each vertex of a triangle to the midpoint of the opposite side.
• The centroid has a consistent property - it's located at the average position of all the points in the triangle, making it a natural balance point.
• In geometry problems, finding the centroid is a common task that involves understanding both the coordinate positions of the vertices and applying a straightforward formula.

Finding centroids is essential in applications across different fields such as engineering, architecture, and even robotics. In these fields, knowing how a shape will balance or pivot can be crucial for designs and functionality.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, merges algebra and geometry to solve problems about geometric figures using coordinates. In coordinate geometry, each point in the plane is described using an ordered pair \(x, y\), and equations like lines and curves can be represented algebraically.

For instance, in our exercise, the line \(x+y=3\) represents a straight line in the coordinate plane:

• This line forms the boundary of the triangular region, while the x-axis and y-axis serve as the other two boundaries.
• Identifying points of intersection, such as where the line meets the axes, helps determine the vertices of the triangle.
• The vertices provide essential data used in the centroid formula – which creatively combines geometry with algebra by averaging the x and y coordinates.

Combining these coordinates to identify points and solve geometric problems is what makes coordinate geometry a powerful and versatile tool.

First Quadrant Geometry

First quadrant geometry focuses on understanding figures located within the first quadrant of a Cartesian coordinate system.
The first quadrant is defined by both the x and y coordinates being positive, which simplifies many calculations and visualizations.

• The concept of working exclusively in this quadrant can make problems easier since we don't have to worry about negative coordinates or mirrored symmetry used in the other quadrants.
• In our triangle example formed by the line \(x+y=3\), only positive values of x and y are used to describe the locations of vertices, which makes calculations straightforward.
• First quadrant geometry also encourages practical thinking, as many real-life applications focus on components located in one "positive" area, like in city block layouts or park designs.

Focusing on the first quadrant aids thestudents in developing a clear, visual understanding of geometry within a familiar and manageable space.