Find the area of the triangle ABC if A (-4; 1), B (1; 0) and C (2; 4).

univerkov | August 10, 2021 | education

Let’s find the lengths of the sides of this triangle ABC, and then use Heron’s formula for the area of the triangle.

| AB | = √ ((- 4 – 1) ^ 2 + (1 – 0) ^ 2) = √ ((- 5) ^ 2 + 1 ^ 2) = √ (25 + 1) = √26;

| BC | = √ ((2 – 1) ^ 2 + (4 – 0) ^ 2) = √ (1 ^ 2 + 4 ^ 2) = √ (1 + 16) = √17;

| FC | = √ ((- 4 – 2) ^ 2 + (1 – 4) ^ 2) = √ (6 ^ 2 + 3 ^ 2) = √ (36 + 9) = √45.

We find the half-measure p of this triangle:

p = (√26 + √17 + √45) / 2.

Find the area of this triangle:

S = √ (p * (p – √26) * (p – √17) * (p – √45)) = √ (((√26 + √17 + √45) / 2) * ((-√26 + √17 + √45) / 2) * ((√26 – √17 + √45) / 2) * ((√26 + √17 – √45) / 2)) = 0.25 * √ ((√26 + √17 + √45) * (-√26 + √17 + √45) * (√26 – √17 + √45) * (√26 + √17 – √45)).

Answer: 0.25 * √ ((√26 + √17 + √45) * (-√26 + √17 + √45) * (√26 – √17 + √45) * (√26 + √17 – √45) ).

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