Question

defination of area of triangle with known coordinates of vertices

Answer 1

MATH ON ROUGH SHEETS

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How to find area of triangle given three vertices?

August 5, 2012 by Jashan 

 

We can find Area of triangle using formula 12× base × height if we know the length of base and height of triangle.

 

We can also find area of triangle usingHeron's formula if we know the length of three sides of triangle.

But, how can we find area of triangle if we know only the coordinates of vertices of triangle. If, we know the vertices of triangle then we can definitely use distance formula to find the length of all the sides which can enable us to use Heron's formula to find area of triangle. But, this will become too much lengthy and tedious.

 

We have a formula which can be directly used on the vertices of triangle to find its area.

If, (x1, x2), (x2, y2) and (x3, y3) are the coordinates of vertices of triangle then

 

Area of Triangle = 12(x1(y2−y3)+x2(y3−y1)+x3(y1−y2))

 

Now, we can easily derive this formula using a small diagram shown below.



 

Suppose, we have a △ABC as shown in the diagram and we want to find its area.

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

We draw perpendiculars AP, BQ and CR to x-axis.

 

Area of △ABC = Area of Trapezium ABQP + Area of Trapezium BCRQ - Area of Trapezium ACRP

 

⇒ Area of △ABC=12(y1+y2)(x1−x2)

+12(y1+y3)(x3−x1)

−12(y2+y3)(x3−x2)=12(x1(y2−y3)+x2(y3−y1)+x3(y1−y2))

 

 

Example: Find area of triangle whose vertices are (1, 1), (2, 3) and (4, 5)

 

Solution:

We have (x1, y1) = (1, 1), (x2, y2) = (2, 3) and (x3, y3)