. Find The Area Of The Given Figure. 10.2,3.9,1.5

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Introduction

In mathematics, finding the area of a figure is a fundamental concept that involves calculating the amount of space inside a shape. The area of a figure can be determined using various formulas, depending on the type of shape. In this article, we will explore how to find the area of a given figure using the coordinates of its vertices.

Understanding the Problem

To find the area of a figure, we need to know the coordinates of its vertices. The given figure has three vertices with coordinates (10.2, 3.9), (1.5, 3.9), and (1.5, 1.5). We will use these coordinates to find the area of the figure.

The Shoelace Formula

The Shoelace formula is a popular method for finding the area of a simple polygon whose vertices are given by their coordinates in the plane. The formula is as follows:

Area = (1/2) * |(x1y2 + x2y3 + x3y1) - (y1x2 + y2x3 + y3x1)|

where (x1, y1), (x2, y2), and (x3, y3) are the coordinates of the vertices of the polygon.

Applying the Shoelace Formula

Let's apply the Shoelace formula to the given figure. We have the following coordinates:

(x1, y1) = (10.2, 3.9) (x2, y2) = (1.5, 3.9) (x3, y3) = (1.5, 1.5)

Substituting these values into the Shoelace formula, we get:

Area = (1/2) * |(10.23.9 + 1.51.5 + 1.53.9) - (3.91.5 + 3.91.5 + 1.510.2)| Area = (1/2) * |(39.78 + 2.25 + 5.85) - (5.85 + 5.85 + 15.3)| Area = (1/2) * |(47.88) - (27.0)| Area = (1/2) * |20.88| Area = 10.44

Conclusion

In this article, we used the Shoelace formula to find the area of a given figure with coordinates (10.2, 3.9), (1.5, 3.9), and (1.5, 1.5). The area of the figure is 10.44 square units. The Shoelace formula is a powerful tool for finding the area of simple polygons, and it can be applied to a wide range of problems in mathematics and computer science.

Real-World Applications

The area of a figure has many real-world applications, including:

  • Geometry and Trigonometry: The area of a figure is a fundamental concept in geometry and trigonometry, and it is used to solve problems involving the area of triangles, quadrilaterals, and other polygons.
  • Computer Graphics: The area of a figure is used in computer graphics to determine the size and shape of objects on a screen.
  • Engineering: The area of a figure is used in engineering to calculate the stress and strain on materials, and to design structures such as bridges and buildings.
  • Science: The area of a figure is used in science to calculate the surface area of objects, and to determine the volume of objects.

Tips and Tricks

Here are some tips and tricks for finding the area of a figure:

  • Use the Shoelace formula: The Shoelace formula is a powerful tool for finding the area of simple polygons, and it can be applied to a wide range of problems.
  • Check your work: Always check your work to make sure that you have found the correct area.
  • Use a calculator: A calculator can be a big help when finding the area of a figure, especially if you are dealing with complex calculations.
  • Practice, practice, practice: The more you practice finding the area of a figure, the more comfortable you will become with the formulas and techniques.

Common Mistakes

Here are some common mistakes to avoid when finding the area of a figure:

  • Not using the correct formula: Make sure to use the correct formula for the type of figure you are working with.
  • Not checking your work: Always check your work to make sure that you have found the correct area.
  • Not using a calculator: A calculator can be a big help when finding the area of a figure, especially if you are dealing with complex calculations.
  • Not practicing: The more you practice finding the area of a figure, the more comfortable you will become with the formulas and techniques.

Conclusion

In conclusion, finding the area of a figure is a fundamental concept in mathematics that involves calculating the amount of space inside a shape. The Shoelace formula is a powerful tool for finding the area of simple polygons, and it can be applied to a wide range of problems in mathematics and computer science. By following the tips and tricks outlined in this article, and by avoiding common mistakes, you can become proficient in finding the area of a figure.

Q: What is the Shoelace formula?

A: The Shoelace formula is a method for finding the area of a simple polygon whose vertices are given by their coordinates in the plane. It is a popular method for finding the area of a figure and is widely used in mathematics and computer science.

Q: How do I apply the Shoelace formula?

A: To apply the Shoelace formula, you need to substitute the coordinates of the vertices of the polygon into the formula. The formula is as follows:

Area = (1/2) * |(x1y2 + x2y3 + x3y1) - (y1x2 + y2x3 + y3x1)|

where (x1, y1), (x2, y2), and (x3, y3) are the coordinates of the vertices of the polygon.

Q: What are the coordinates of the vertices?

A: The coordinates of the vertices are the x and y values that define the position of each vertex of the polygon. For example, if the vertex is located at (3, 4), the coordinates are (3, 4).

Q: How do I find the area of a triangle?

A: To find the area of a triangle, you can use the Shoelace formula. However, there is a simpler formula for finding the area of a triangle:

Area = (1/2) * base * height

where base is the length of the base of the triangle and height is the height of the triangle.

Q: How do I find the area of a rectangle?

A: To find the area of a rectangle, you can use the formula:

Area = length * width

where length is the length of the rectangle and width is the width of the rectangle.

Q: What is the difference between the Shoelace formula and the formula for finding the area of a triangle?

A: The Shoelace formula is a general formula for finding the area of a simple polygon, while the formula for finding the area of a triangle is a special case of the Shoelace formula. The Shoelace formula can be used to find the area of any simple polygon, while the formula for finding the area of a triangle is only used for triangles.

Q: Can I use the Shoelace formula to find the area of a complex polygon?

A: The Shoelace formula is only used for simple polygons, which are polygons that do not intersect themselves. If you have a complex polygon, you may need to use a different method to find its area.

Q: How do I check my work when finding the area of a figure?

A: To check your work, you can use a calculator to plug in the values and see if you get the same answer. You can also use a graphing calculator to visualize the polygon and see if the area makes sense.

Q: What are some common mistakes to avoid when finding the area of a figure?

A: Some common mistakes to avoid when finding the area of a figure include:

  • Not using the correct formula for the type of figure you are working with
  • Not checking your work to make sure that you have found the correct area
  • Not using a calculator to help with complex calculations
  • Not practicing to become proficient in finding the area of a figure

Q: How can I practice finding the area of a figure?

A: You can practice finding the area of a figure by working through problems and exercises in a textbook or online resource. You can also try creating your own problems and solving them to test your skills.

Q: What are some real-world applications of finding the area of a figure?

A: Finding the area of a figure has many real-world applications, including:

  • Geometry and trigonometry
  • Computer graphics
  • Engineering
  • Science

Q: How can I use the area of a figure in real-world applications?

A: You can use the area of a figure in real-world applications by applying the formulas and techniques you have learned to solve problems and make calculations. For example, you can use the area of a triangle to calculate the height of a building, or the area of a rectangle to calculate the size of a room.