Find the area of a triangle ABC if A (3; 0; 0), B (0; -4; 0), C (0; 0; 1).

1. Let’s mark the points at the given coordinates. Got a triangle ABC.

The area S will be calculated by Heron’s formula: S = √p * (p – a) * (p – b) * (p – c), where a, b, c are the sides of the triangle, and h = (a + b + c ): 2.

2. The length of the side AB is calculated as the hypotenuse of the triangle, one leg of which is equal to the length of the projection of the segment AB on the abscissa axis, and the other on the ordinate axis:

AB = √ 4² + 3² = √25 = 5.

We also define the sides of the AC and BC:

AC = √3² + 1² = √10 = 3.16,

CA = √4² + 1² = √ 17 = 4.12

The semi-perimeter p is (5 + 3.16 + 4.12): 2 = 12.28: 2 = 6.14.

3.S treug = √6.14 * (6.14 – 5) * (6.14 – 3.16) * (6.14 – 4.12) = √ 6.14 * 1.14 * 2.98 * 2.02 = √ 42 = 6.5.

Answer: The area is 6.5