Find the Area of a Triangle

Find the area of a triangle whose vertices are defined by the points A(1, 3), B(–2, 1), and C(4, 2).

STEP 1:   Construct the triangle using the given 3 coordinate points.

slope m and y-intercept b

You have labeled the three sides of the triangle as a, b, and c. Since the coordinates of the three vertices are known, you can find the distance between these vertices using the distance formula.

Distance formula: If two points (x1, y1) and (x2, y2) are given, the distance between these points is given by the formula:

distance formula
Find the distance between the vertices using the formula.

using the distance formula

Now, you have obtained the lengths of the sides of the triangle. However, the height of the triangle cannot be determined. So, it is not possible to find the area of the triangle using the basic formula:

area of triangle formula

STEP 2: Recollect the Heron’s formula for triangles. The formula helps us to find the area of a triangle if all its sides are known.

Heron's formula
In the formula, the variable s represent the semi-perimeter of the triangle, and variables a, b, c are the side lengths. To find s, you need to use another relation:

s formula

Substitute for the sides.

using the s formula
The semi-perimeter is 6.45 units.

STEP 3:    The final step is to replace all the variables in the Heron’s formula with the known values.

using the heron formula
You finally get the area of the triangle as 4.57 square units.

Comments for Find the Area of a Triangle

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Mar 17, 2016
by: HAshim LAwandi TELa

I humbly appreciate this gesture, keep it up! #Hashematics

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