What is the area of the triangle joining the points (3, 4), (2, 1) and (6, 0)Is there an easier way than using Heron's formula

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justaguide eNotes educator| Certified Educator

The area of the triangle formed by joining the points (6, 0), (2, 1) and (3, 4) has to be determined. Instead of using Heron's formula, the formula area = (1/2)*base*height can be used by calculating the base and the height.

Let the line joining the points (3, 4) and (2, 1) be the base. The length of the line joining them is `sqrt((3 - 2)^2 + (4 - 1)^2) = sqrt(1 + 9) = sqrt 10` . The equation of the line is `(y - 1)/(x - 2) = 3/1`

=> `y - 1 = 3x - 6`

=> `3x - y - 5 = 0`

The height of the triangle is the perpendicular distance of the point (6, 0) from the line 3x - y - 5 = 0. This is equal to `h = |6*3 - 0 - 5|/sqrt(3^2 + 1^2) = 13/sqrt 10`

The area of the triangle is `(1/2)(13/sqrt 10)(sqrt 10) = 6.5`

The area of the triangle formed by the points (3, 4), (2, 1) and (6, 0) is 6.5