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Coordinate Geometry

Distance Formula

Let us consider a triangle whose base is parallel to x-axis.

Can you find its area?

Yes, it is a simple question to us as we know that area of triangle is given by the formula

Area

From the figure, it is clear that height of the triangle is 3 units and base is also 3 units.

Area of triangle ABC

square units

Now, consider the following figure

Now, can we calculate the area of ΔDEF? Here, we do not know the base and height. It is very difficult to find the base and height of ΔDEF but we can find the vertices of ΔDEF very easily.

The co-ordinates of D are (2, 5).

The co-ordinates of E are (5, 2).

The co-ordinates of F are (4, 7).

We can calculate the area of ΔDEF by a formula which involves the vertices of a triangle.

Let us derive that formula by considering any triangle, say ΔPQR, such that (x1, y1), (x2, y2), and (x3, y3) are the coordinates of the vertices P, Q and R respectively. 

Here, line segments PA, QB and RC are the perpendiculars to X-axis from the vertices P, Q and R respectively. Therefore, PA || QB || RC and hence, quadrilaterals PQBA, PACR and QBCR are trapeziums.

From the figure, it can be seen that

Area of ΔPQR = Area of trapezium PQBA + Area of trapezium PACR – QBCR    

We know that 

Area of trapezium = (Sum of parallel sides × Perpendicular distance between parallel sides)

Therefore,

Area of ΔPQR = (QB + PA)BA + (PA + RC)AC – (QB + RC)BC

                       

∴ Area of triangle

 

Now, let us find the area of ΔDEF using this formula.

Area of triangle DEF

Area of ΔDEF = 6 square units

In this way, we can calculate the area of a triangle in a coordinate plane by using this formula.

Can we have a triangle with area 0 square units? Let us see this.

Let us find the area of a triangle formed by the vertices (1, 4), (−1, 1), and (3, 7).

Now, area

The area of the triangle is 0. What does it mean?

It means that the three given points are collinear.

Thus, “if area of a triangle is zero, then its vertices will be collinear”.

Let us solve some more examples.

Example 1:

Find the area of a triangle whose vertice…

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