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Magazine Chapter 9: Problem 26
Find the centroid and area of the figure with the given vertices. $$(-3,-2),(-2,4),(6,4),(7,-2)$$
Short Answer
Expert verified
Answer: The coordinates of the centroid are $(\frac{11}{6}, 1)$ and the area of the figure is $20$.
Step by step solution
01
Identify the vertices of the figure
The given vertices of the figure are: $$(-3,-2),(-2,4),(6,4),(7,-2)$$
02
Divide the figure into triangles
By observing the figure or plotting the points, we can notice that the figure is a parallelogram. We can divide it into two triangles: Triangle 1 with vertices $$(-3,-2),(-2,4),(6,4)$$ and Triangle 2 with vertices $$(-3,-2),(6,4),(7,-2)$$.
03
Calculate the area of each triangle
We'll use the Shoelace Theorem to calculate the area of each triangle.
For Triangle 1:
$$A_1 = \frac{1}{2} |(-3)(4) + (-2)(4) + (6)(-2)| = \frac{1}{2} \cdot 20 = 10$$
For Triangle 2:
$$A_2 = \frac{1}{2} |(-3)(4) + (6)(-2) + (7)(-2)| = \frac{1}{2} \cdot 20 = 10$$
04
Calculate the total area of the figure
Add the areas of both triangles to find the total area of the figure.
$$A = A_1 + A_2 = 10 + 10 = 20$$
05
Find the centroid of each triangle
We will average the x-coordinates and y-coordinates of the vertices for each triangle.
Centroid of Triangle 1:
$$(C_{1x}, C_{1y}) = \left(\frac{-3 + (-2) + 6}{3}, \frac{-2 + 4 + 4}{3}\right) = (1/3, 2)$$
Centroid of Triangle 2:
$$(C_{2x}, C_{2y}) = \left(\frac{-3 + 6 + 7}{3}, \frac{-2 + 4 + (-2)}{3}\right) = (10/3, 0)$$
06
Find the centroid of the figure
We will average the centroids of both triangles to find the centroid of the figure.
$$(C_x, C_y) = \left(\frac{1/3 + 10/3}{2}, \frac{2 + 0}{2}\right) = \left(\frac{11}{6}, 1\right)$$
So, the centroid of the figure is $$(\frac{11}{6}, 1)$$ and the area of the figure is $$20$$.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Centroid Calculation
Finding the centroid of a shape involves determining its "center of mass," akin to balancing the shape on a pinpoint. For any polygon, we can calculate its centroid by averaging the x- and y-coordinates of its vertices, especially if we divide the polygon into simpler shapes, like triangles.
- For each triangle within the polygon, we calculate its centroid (called the geometric center) by averaging its vertex coordinates.
For example, for a triangle with vertices \((-3, -2), (-2, 4), (6, 4)\), the centroid in x is calculated as \(\frac{-3 + (-2) + 6}{3}\) and in y as \(\frac{-2 + 4 + 4}{3}\). - Once this is done for each triangle, we find the overall centroid by averaging the centroids of the individual triangles.
Area of a Polygon
The area of a polygon is the space it occupies on a plane. Calculating the area involves understanding the geometry of the shape and often, for irregular polygons, subdividing it into manageable shapes.
- For instance, consider a polygon resembling a parallelogram, divided into two triangles for easier area calculation.
We can calculate each triangle's area separately and sum them up to get the total area. - Using the vertices of the triangles, apply geometric formulas to find the individual areas. Summing these gives the polygon's total area.
Shoelace Theorem
The Shoelace Theorem is a nifty technique for calculating the area of a polygon when its vertices are known. The term comes from the crisscross pattern, resembling a laced shoe, made by multiplying and summing specific sets of coordinates.
Here's how it works:
Here's how it works:
- List all vertices in order, creating a loop from the last back to the first. For example, in a triangle with vertices \((-3, -2), (-2, 4), (6, 4)\), cycle back to \((-3, -2)\).
- Multiply diagonally across coordinates in one direction, add up these products.
- Multiply diagonally in the opposite direction, subtract this sum from the first sum.
- The final result is halved (absolute value) to give the area.
