# Exploring the Formula for Finding the Area of Triangle in Coordinate Geometry

Have you ever wondered how to find the area of a triangle in coordinate geometry? If so, you're in the right place! It is very important for you to get your basics write. This is only possible with professional assistance- Cuemath!

In this blog, we will explore the formula for searching the area of a triangle in coordinate geometry and how it can be used to solve problems. We will also discuss how to find the perimeter of a triangle with vertices.

## Area of Triangle in Coordinate Geometry

To find the area in coordinate geometry, we need to use the coordinates of the vertices.

The formula to get the size of a triangle with vertices (x1, y1), (x2, y2), and (x3, y3) is:

Area = 1/2 * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|

Let us break down the formula step by step. First, we take the absolute value of the sum of three terms. Each term is the product of one of the x-coordinates and the difference between the y-coordinates of two vertices. The factor 1/2 is then multiplied to get the final area.

## Area of Triangle in Coordinate Geometry with 3 Points Formula

Another way to get the area of a triangle in coordinate geometry is by using the distance formula. If we have the coordinates of the vertices (x1, y1), (x2, y2), and (x3, y3), we can use the distance formula to get the length of the sides of the triangle. Let us call these lengths a, b, and c. Then, we can use Heron's formula:

Area = √(s(s-a)(s-b)(s-c))

where s is the semi-perimeter of the triangle, which is half the perimeter.

## Find the triangle's Perimeter with Vertices

To find the perimeter of a triangle with vertices, we simply need to find the distance between each pair of vertices and add them together. If we have the coordinates of the vertices (x1, y1), (x2, y2), and (x3, y3), the distance between (x1, y1) and (x2, y2) is:

Distance = √((x2 - x1)^2 + (y2 - y1)^2)

We can find the other two distances similarly and add them together to get the perimeter of the triangle.

In addition to the formula discussed above, there are other methods for finding the area of a triangle in coordinate geometry. One such way is to use the determinant of a matrix. The determinant of a 2x2 matrix can be used to find the triangle's area in the coordinate plane. To use this method, we need to find the determinant of a 3x3 matrix, where the first two rows represent the x and y coordinates of the three points, and the last row is filled with ones.

The formula to get the determinant of a 3x3 matrix is:

| a b c | | d e f | | g h i |

Det = a(ei - fh) - b(di - fg) + c(dh - eg)

To find the triangle's area using this method, we can form a 3x3 matrix using the coordinates of the three points and then find the determinant of the matrix. The absolute value of the determinant divided by 2 gives us the area of the triangle.

Another useful formula in coordinate geometry is the formula to find the perimeter of a triangle with vertices (x1, y1), (x2, y2), and (x3, y3). The formula is:

Perimeter = AB + BC + AC

where AB, BC, and AC are the distances between the vertices.

The distance between two points can be searched using the distance formula:

Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Using the distance formula, we can get the lengths of each side and then add them together to find the perimeter.

The area can be found using the formula 1/2 * base * height or the determinant of a 3x3 matrix. Additionally, the perimeter of a triangle can be found using the distance formula and the formula for the sum of the side lengths. By understanding these formulas, we can quickly solve various problems in coordinate geometry.

The formula to get the area of a triangle in coordinate geometry is a valuable tool for solving problems in mathematics. It can also be used in real-world applications such as calculating the area of a piece of land or determining the amount of material needed for a construction project. To further improve your skills in coordinate geometry, Cuemath offers online classes with expert math tutors who can guide you through the concepts step by step. Sign up today and start learning in fun and engaging way!

### Conclusion

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### FAQs

### 1. What is the difference between the formula to find a triangle's area in coordinate geometry and the get the area of a triangle in regular geometry?

The formula to get the area of a triangle in regular geometry is (1/2)bh, where b is the base of the triangle and h is the height. In coordinate geometry, we use the coordinates of the vertices of the triangle to get the length of its sides and then use the Heron's formula or the formula for the determinant of a matrix to find its area.

### 2. Can I find the area of a triangle in coordinate geometry with only two points?

No, you need three points to find the area of a triangle in coordinate geometry. Two points only give you a line segment, not a triangle.

### 3. What is the practical application of finding the triangle's area in coordinate geometry?

The area of a triangle is a fundamental concept used in many real-world applications, such as architecture, engineering, and surveying. For example, if you want to build a roof over a triangular-shaped house, you need to know the area of the triangle to estimate the number of materials you need to cover the top.