What Is The Area Of A Triangle With Vertices At $(-2,-1$\], $(4,-1$\], $(6,5$\]?A. 6 Square Units B. 9 Square Units C. 18 Square Units D. 36 Square Units

Feb 27, 2025 by ADMIN 159 views

**What is the Area of a Triangle with Given Vertices?**

Introduction

In geometry, the area of a triangle can be calculated using various methods, including the Shoelace formula, Heron's formula, and the formula for the area of a triangle given its vertices. In this article, we will explore the latter method and use it to find the area of a triangle with vertices at $(-2,-1)$ , $(4,-1)$ , and $(6,5)$ .

The Formula for the Area of a Triangle Given its Vertices

The formula for the area of a triangle given its vertices $(x_1, y_1)$ , $(x_2, y_2)$ , and $(x_3, y_3)$ is:

A = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|

This formula is derived from the shoelace formula and is a simple and efficient way to calculate the area of a triangle given its vertices.

Calculating the Area of the Triangle

Now that we have the formula, let's calculate the area of the triangle with vertices at $(-2,-1)$ , $(4,-1)$ , and $(6,5)$ . We will plug these values into the formula and simplify to find the area.

First, we identify the coordinates of the vertices:

$(x_1, y_1) = (-2, -1)$
$(x_2, y_2) = (4, -1)$
$(x_3, y_3) = (6, 5)$

Next, we plug these values into the formula:

A = \frac{1}{2} |-2(-1 - 5) + 4(5 - (-1)) + 6((-1) - (-1))|

Simplifying the expression inside the absolute value, we get:

A = \frac{1}{2} |-2(-6) + 4(6) + 6(0)|

A = \frac{1}{2} |12 + 24 + 0|

A = \frac{1}{2} |36|

A = \frac{1}{2} \times 36

A = 18

Conclusion

The area of the triangle with vertices at $(-2,-1)$ , $(4,-1)$ , and $(6,5)$ is 18 square units. This is the final answer to the problem.

Answer

The correct answer is C. 18 square units.

Why is this Formula Important?

The formula for the area of a triangle given its vertices is an important concept in geometry and is used in a variety of applications, including:

Calculating the area of a triangle in a coordinate plane
Finding the area of a triangle in a real-world scenario, such as calculating the area of a triangle-shaped region on a map
Using the formula as a building block for more complex geometric calculations

Real-World Applications

The formula for the area of a triangle given its vertices has many real-world applications, including:

Surveying: When surveying a piece of land, it is often necessary to calculate the area of a triangle-shaped region. This formula can be used to do so.
Architecture: When designing a building, architects may need to calculate the area of a triangle-shaped region, such as a triangular roof or a triangular section of a wall.
Engineering: Engineers may use this formula to calculate the area of a triangle-shaped region in a mechanical system, such as a triangle-shaped gear or a triangle-shaped component.

Conclusion

Q: What is the formula for the area of a triangle given its vertices?

A: The formula for the area of a triangle given its vertices $(x_1, y_1)$ , $(x_2, y_2)$ , and $(x_3, y_3)$ is:

A = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|

Q: How do I use the formula to calculate the area of a triangle?

A: To use the formula, simply plug in the coordinates of the vertices into the formula and simplify to find the area.

Q: What if the vertices are not given in the standard (x, y) format?

A: If the vertices are given in a different format, such as (x, y, z) or (r, θ), you will need to convert them to the standard (x, y) format before using the formula.

Q: Can I use the formula to calculate the area of a triangle with negative coordinates?

A: Yes, the formula can be used to calculate the area of a triangle with negative coordinates. Simply plug in the coordinates and simplify to find the area.

Q: What if the triangle is not a right triangle?

A: The formula can be used to calculate the area of any triangle, regardless of whether it is a right triangle or not.

Q: Can I use the formula to calculate the area of a triangle with complex coordinates?

A: Yes, the formula can be used to calculate the area of a triangle with complex coordinates. However, you will need to use the complex number arithmetic rules to simplify the expression.

Q: Is there a way to simplify the formula for a triangle with specific coordinates?

A: Yes, there are ways to simplify the formula for a triangle with specific coordinates. For example, if the triangle has vertices at (0, 0), (a, 0), and (0, b), the formula can be simplified to:

A = \frac{1}{2} ab

Q: Can I use the formula to calculate the area of a triangle with coordinates that are not integers?

A: Yes, the formula can be used to calculate the area of a triangle with coordinates that are not integers. Simply plug in the coordinates and simplify to find the area.

Q: Is there a way to visualize the area of a triangle using the formula?

A: Yes, there are ways to visualize the area of a triangle using the formula. For example, you can use a graphing calculator or a computer program to plot the triangle and calculate its area.

Q: Can I use the formula to calculate the area of a triangle with coordinates that are in a different unit system?

A: Yes, the formula can be used to calculate the area of a triangle with coordinates that are in a different unit system. Simply plug in the coordinates and simplify to find the area, and be sure to use the correct unit conversions.

Conclusion

In conclusion, the formula for the area of a triangle given its vertices is a powerful tool for calculating the area of a triangle. By understanding the formula and how to use it, you can solve problems in geometry and other fields.