Problem 29 - Basic Technical Mathematics with Calculus

Question

Find the coordinates of the centroids of the given figures. Each region is covered by a thin, flat plate. A sailboat has a right-triangular sail with a horizontal base \(3.0 \mathrm{m}\) long and a vertical side of \(4.5 \mathrm{m}\) high. Where is the centroid of the sail?

Step-by-Step Solution

Answer

The centroid of the sail is at \((1.0, 1.5)\).

Step 1: Understand the problem

We need to find the centroid of a right triangle. The right triangle in question has a horizontal base of \(3.0 \mathrm{m}\) and a vertical height of \(4.5 \mathrm{m}\). The centroid of a right triangle is the point that divides each median in the ratio 2:1.

Step 2: Recall the formula for the centroid

In a right triangle, the centroid \((x_c, y_c)\) can be found using the formula: \(x_c = \frac{x_1 + x_2 + x_3}{3}\) and \(y_c = \frac{y_1 + y_2 + y_3}{3}\), where \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) are the vertices of the triangle.

Step 3: Assign coordinates to the vertices

Assume the base of the triangle is on the x-axis starting from the origin. The coordinates are: \((0, 0)\) for the origin, \((3.0, 0)\) for the other end of the base, and \((0, 4.5)\) for the vertex on the y-axis.

Step 4: Apply the formula for x-coordinates

Substitute the x-coordinates of the vertices into the centroid formula: \(x_c = \frac{0 + 3.0 + 0}{3} = \frac{3.0}{3} = 1.0\).

Step 5: Apply the formula for y-coordinates

Substitute the y-coordinates of the vertices into the centroid formula: \(y_c = \frac{0 + 0 + 4.5}{3} = \frac{4.5}{3} = 1.5\).

Step 6: Conclusion

The centroid of the right triangular sail is at the point \((1.0, 1.5)\).

Key Concepts

Right triangle centroid formula

When dealing with a right triangle, finding the centroid is an interesting task due to the unique properties of these triangles. The centroid, or the geometric center, is the point where all the medians of the triangle intersect. Specifically, for a right triangle, the centroid divides each of the medians in a 2:1 ratio, starting from the right angle to the hypotenuse.

To find the centroid of a right triangle, there is a formula that simplifies the process. By using the coordinates of the vertices of the triangle, the centroid can be calculated using the formulas:

• For the x-coordinate of the centroid: \(x_c = \frac{x_1 + x_2 + x_3}{3}\)
• For the y-coordinate of the centroid: \(y_c = \frac{y_1 + y_2 + y_3}{3}\)

These formulas effectively average the x and y coordinates of the vertices, providing the location of the centroid. It's fascinating how this straightforward mathematical approach gives such vital information about the geometry of the triangle.

Coordinates of vertices

In geometry, vertices are the points where two or more lines meet. In the case of triangles, vertices are crucial because they define the shape and dimensions of the figure. To find the centroid of a triangle, specifically a right triangle, understanding and assigning coordinates to these vertices is a primary step.

For a right triangle positioned in the coordinate plane:

• Choose a logical position. Here, the base of the triangle can be laid on the x-axis. This often makes calculations easier.
• For instance, if we have a base of 3.0 meters, the ends of the base can be at \(0, 0\) and \(3.0, 0\).
• The vertex at the height can be at \(0, 4.5\).

By following this approach, setting coordinates facilitates more straightforward calculations, providing a clear path to finding the centroid.

Geometric centroid

The centroid of a shape is often referred to as its geometric center. It is a crucial concept in geometry, as it represents a point of balance or gravity center for the shape. In simple terms, for a triangle like the right triangular sail given here, the centroid is the point at which the triangle would balance perfectly if it were made of a uniform material.

For a right triangle, the geometric centroid has a special relation with the sides and vertices of the triangle. It is always located inside the triangle, lying along each median. This special property makes the centroid an invaluable tool for construction, engineering, and design purposes.

Interestingly, while every triangle has a centroid, the right triangle's centroid can be located swiftly using the vertex coordinate averaging formula. This gives powerful insights, not only in mathematics but also in practical applications where balance and symmetry are needed.

Centroid calculation

Calculating the centroid of a right triangle involves a few organized steps. By understanding and applying the mathematical concepts and coordinates of the vertices, the process can be easy and straightforward.

Here's a concise guide to calculating the centroid:

• Identify the vertices of your triangle. In our case, these were \(0,0\), \(3.0,0\), and \(0,4.5\).
• Use the centroid formula for both x and y coordinates:
  • For the **x-coordinate**, calculate: \(x_c = \frac{0 + 3.0 + 0}{3} = 1.0\)
  • For the **y-coordinate**, calculate: \(y_c = \frac{0 + 0 + 4.5}{3} = 1.5\)

Once completed, these calculations provide us the centroid at \(1.0, 1.5\). This point is key because it divides all the medians from the vertices to the opposite sides into two segments, maintaining a specific 2:1 ratio. This central point is fascinating in how it distills the triangle's shape into a single, crucially informative location.