Find S of a triangle whose vertices have coordinates (1; 1), (4; 4), (5; 1).

| April 4, 2021 | education

As you know, the distance between points given by their coordinates is calculated by the formula: a = √ (x1 – x2) ² + (y1 – y2) ².
Find the length of the first side of the triangle by substituting the coordinates of the vertices into the formula: a = √ (1 – 4) ² + (1 – 4) ² = √9 + 9 = √18 = 4.2.
Find the length of the second side of the triangle by substituting the coordinates of the vertices into the formula: b = √ (1 – 5) ² + (1 – 1) ² = √16 + 0 = √16 = 4.
Find the length of the third side of the triangle by substituting the coordinates of the vertices into the formula: с = √ (4 – 5) ² + (4 – 1) ² = √1 + 9 = √10 = 3.2.
Find the area of ​​the triangle using Heron’s formula.

Find the semiperimeter of a given triangle by the formula: p = (a + b + c) / 2 = (4.2 + 4 + 3.2) / 2 = 11.4 / 2 = 5.7.
We now calculate the area of ​​a given triangle using Heron’s formula: S = √ (p * (p – a) * (p – b) * (p – c) = √5.7 * (5.7 – 4.2) * ( 5.7 – 4) * (5.7 – 3.2) = √5.7 * 1.5 * 1.7 * 2.5 = √36.3 = 6.03.
Answer: The area of ​​the triangle is 6.03.



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