Coordinate geometry is a branch of math that combines the geometry and algebra. coordinate geometry involves geometric figures using algebraic concepts and techniques. The fundamental of coordinate geometry is how to calculate the area of different shapes, such as triangles. In this article, we will provide the concept of finding the area of triangle formula in coordinate geometry and discuss various methods to find this.

## Definition of a Triangle

A triangle is a polygon with three angles and three sides. It consists of three vertices and three line segments that connect these vertices. The vertices of a triangle can be represented as ordered pairs of numbers (x, y), where x represents the horizontal position and y represents the vertical position.

## What Is Area of Triangle Formula in Coordinate Geometry?

The area of triangle in coordinate geometry is calculated by the formula (1/2) |x_{1}(y_{2} − y_{3}) + x_{2}(y_{3} − y_{1}) + x_{3}(y_{1} − y_{2})|, where (x_{1}, y_{1}), (x_{2}, y_{2}), and (x_{3}, y_{3}) are the vertices of the triangle.

## What Is the Area of a Triangle in Coordinate Geometry?

The area of triangle in coordinate geometry, Triangle can be calculate or solved by using the coordinates of its vertices. Such as a triangle with vertices A(x1, y1), B(x2, y2), and C(x3, y3). The area of triangle formula in coordinate geometry is as follows:

Area = 0.5 * |x1(y2 – y3) + x2(y3 – y1) + x3(y1 – y2)|

In this formula, | | represents the absolute value and the expression. The determinant of a 3×3 matrix is computed inside the absolute value.

Let us understand the concept of the area of a triangle formula in coordinate geometry.

**The example given below,**

Consider these three points: A (−2, 1), B (3,2), C (1,5). If you draw these three points in the plane, you will notice that they are non-collinear, which means that they can be the vertices of a triangle, as shown below:

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## Formulas for Finding the Area of a Triangle

There are multiple formulas available to find the area of triangle in coordinate geometry. The three different formulas are discussed below.

### Area of Triangle Using Base and Height

Finding the area of triangle by using base and height. The base of a triangle is the length of any sides, and the height is the perpendicular distance from the base to the opposite vertex. The formula for the area using base and height is:

Area = (base * height) / 2

**For Example: **Suppose a triangle ABC. Where B=5 and h=8.

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### Area of Triangle Using Vertices

Finding the area of triangle by using its vertices. Suppose the vertices of a triangle are (x1, y1), (x2, y2), and (x3, y3). The area of triangle formula in coordinate geometry using it’s vertices is :

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

The way to find the area of a triangle when vertices in the coordinate plane is known.

Suppose a triangle PQR, whose coordinates P, Q, and R are given (x_{1}, y_{1}), (x_{2}, y_{2}), (x_{3}, y_{3})……… respectively.

### Area of Triangle Using Determinants

Finding the area of triangle using determinants. By arranging the vertices in a matrix form and calculating the determinant. Then area of triangle formula using determinants is:

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

## Coordinates of Triangle Formula

The coordinates of triangle formula is a formula used to find the area of a triangle given the coordinates of its vertices. The formula is: A = (1/2) |x1(y2 – y3) + x2(y3 – y1) + x3(y1 – y2)|

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## Area of Triangle in Coordinate Geometry Questions

**Question 1: **Find the area of the triangle formed by the vertices a(1, 2), b(4, 5), and c(7, 2).

**Solution: **

To find the area of a triangle formed by three vertices in a coordinate plane, Here we can use the Shoelace formula.

Area = 0.5 * |(x1 * y2 + x2 * y3 + x3 * y1) – (y1 * x2 + y2 * x3 + y3 * x1)|

Area of the triangle formed by the vertices A(1, 2), B(4, 5), and C(7, 2):

x1 = 1, y1 = 2 x2 = 4

y2 = 5 x3 = 7, y3 = 2

Area = 0.5 * |(1 * 5 + 4 * 2 + 7 * 2) – (2 * 4 + 5 * 7 + 2 * 1)|

= 0.5 * |(5 + 8 + 14) – (8 + 35 + 2)|

= 0.5 * |27 – 45|

= 0.5 * |-18|

= 0.5 * 18

= 9

Therefore, the area of the triangle formed by the vertices A(1, 2), B(4, 5), and C(7, 2) is** 9 square units**.

**Question 2:** Find the area of the ∆ABC whose vertices are A(1, 2), B(4, 2) and C(3, 5)?

**Solution:**

By Using formula,

A = (1/2) [x_{1} (y_{2} – y_{3} ) + x_{2} (y_{3 }– y_{1} ) + x_{3}(y_{1 }– y_{2})]

A = (1/2) [1(2 – 5) + 4(5 – 2) + 3(2 – 2)]

A = (1/2) [-3 + 12]= 9/2 square units.

Therefore, the area of a triangle ∆ABC is **9/2 square units.**

**Question 3: **Find the area of triangle whose vertices are (1, –1), (– 4, 6) and (–3, –5).

**Solution:**

Given vertices are A(1, -1), B(-4, 6), and C(-3, -5).

(1 * 6) + (-4 * -5) + (-3 * -1) – ((-1 * -4) + (6 * -3) + (-5 * 1))

= 6 + 20 + 3 – (4 – 18 – 5)

= 29 – 17

= 12

Therefore, the area of triangle is **12 square units.**

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## Frequently Asked Questions on Area of Triangle Formula in Coordinate Geometry

### What Is the Formula to Calculate Area of Triangle in Coordinate Geometry?

The area of triangle formula in coordinate geometry is calculated by (1/2) |x1(y2 − y3) + x2(y3 − y1) + x3(y1 − y2)|.

### How Can I Find the Length of Side of a Triangle Using Coordinates Geometry?

To find the length of a side of a triangle using coordinates Geometrye, you can use the distance formula. The distance formula states that the distance between two points is equal to: √(x2 – x1)^2 + (y2 – y1)^2. Where (x1, y1) and (x2, y2) are the coordinates of the two points.

### How Do You Find the Area of a Triangle With Coordinates?

The area of a triangle can be found using the coordinates of its vertices. The formula for finding the area of a triangle with coordinates is: (1/2) |x1(y2 – y3) + x2(y3 – y1) + x3(y1 – y2)|.

### What Is Area of Triangle Given 3 Points Vectors?

The area of a triangle can be found using the vectors of its vertices. The formula for finding the area of a triangle with vectors is: A = 1/2 |(a – b) × (a – c)|. **Where** A is the area of the triangle and a, b, and c are the vectors of the vertices of the triangle.

### How to Find the Area of a Triangle Worksheet?

Some steps on how to find the area of a triangle worksheet:

- Identify the vertices of the triangle. The vertices are the points where the three sides of the triangle meet.
- Substitute the coordinates of the vertices into the formula for the area of a triangle with coordinates. The formula is: A = (1/2) |x1(y2 – y3) + x2(y3 – y1) + x3(y1 – y2)|.

### What Is Formula for Area of Triangle With 3 Coordinates?

The formula for the area of a triangle with 3 coordinates is: A = (1/2) |(x1)(y2 – y3) + (x2)(y3 – y1) + (x3)(y1 – y2)|, Where (x1, y1), (x2, y2), and (x3, y3) are the coordinates of the triangle’s vertices.

### Perimeter of Triangle Formula in Coordinate Geometry?

Perimeter of triangle is the sum of the lengths of its 3 sides. The lengths of the sides of a triangle can be found using the distance formula in coordinate geometry.

The distance formula states distance between two points, P1(x1, y1) and P2(x2, y2), And the perimeter of triangle formula in coordinate is d = √((x2 – x1)^2 + (y2 – y1)^2).

### How to Use the Area of a Triangle Formula in Coordinate Geometry?

We can use area of triangle formula in coordinate geometry, when we need to know the coordinates of the vertices of the triangle. Once we have the coordinates, we can simply plug them into the formula and simplify.

### What Happens if the Three Points Are Collinear?

The triangle’s area will be 0 if all three points are collinear. This is because a group of collinear points cannot make a triangle.

### There are Another Ways to Find the Area of a Triangle in Coordinate Geometry?

Yes, there are another ways to find the area of a triangle in coordinate geometry.