20.10 - Area of A Triangle

In the previous section, we saw some solved examples related to properties of determinants. In this section, we will see area of triangles.

Some basics can be written in 3 steps:
1. In our earlier coordinate geometry classes, we have seen an expression to find the area of any triangle:

$\text{Area} = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$
(Details here)

2. Now consider the determinant:

$\Delta = \frac{1}{2} \left|\begin{array}{r}      
x_1    &{    y_1    }    &{    1    }    \\
x_2    &{    y_2    }    &{    1    }    \\
x_3    &{    y_3    }    &{    1    }    \\
\end{array}\right|$
• If we expand this determinant along the first column, we will get the same expression as in (1).

3. That means, area of a triangle can be written in "determinant form" also.
• But it is important to take the absolute value of the determinant. This is because, area cannot be a negative quantity.

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Now we will see some solved examples.

Solved example 20.13
Find the area of the triangle whose vertices are (-1.5,2), (-2.5,-2), (4,1.5)
Solution:

◼ Remarks:
• 3: Here we apply two operations:
   ♦ R₁ → R₁ − R₃.
   ♦ R₂ → R₂ − R₃.
4: Here we expand the determinant along C₃.

Solved example 20.14
Find the equation of the line joining A(1,3) and B(0,0) using determinants and find k if D(k,0) is a point such that area of triangle ABD is 3 sq units.
Solution:
Part (i):
1. Let P(x,y) be any point on the line AB.
Then area of the triangle APB = 0. This is because, three collinear points cannot form a triangle.
2. So we can write:

◼ Remarks:
• 2 (magenta color): Here we expand the determinant along the second row.
• 3 (magenta color): Here we use the index (2+3). This is because, '1' is the a₂₃^th element.

3. Since the area is zero, we can write:
-(1/2)(y-3x) = 0
• From this we get: -y + 3x = 0
• Which is same as: y = 3x

4. The line y=3x will pass through both A(1,3) and B(0,0).

Part (ii):

◼ Remarks:
• 2 (magenta color): Here we write $\pm 3$ because, the value of the determinant can be either +3 or -3. We can ignore the -ve sign only when we write it as an area.
• 3 (magenta color): Here we expand the determinant along the second row.
• 3 (magenta color): Here we use the index (2+3). This is because, '1' is the a₂₃^th element.

◼ We can write:
• Area of the triangle formed by the three points A(1,3), B(0,0) and D(2,0) is 3 units.
• Area of the triangle formed by the three points A(1,3), B(0,0) and D(-2,0) is also 3 units. 

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The link below gives a few more solved examples:

Exercise 20.3

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In the next section, we will see Minors and Cofactors.

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