Question

Докажите что треугольник АВС прямоугольный если А(-3;2;1), В(1;1;2), С(7;20;-3;)

Answer 1

Находим длины сторон заданного треугольника.
Расстояние между точками:
d = √((х₂ - х₁ )² + (у₂ - у₁ )² + (z₂ – z₁ )²)
Подставим координаты точек и получаем:
      АВ                    ВС                      АС
    √18                   √422                   √440
4.2426407         20.542639            20.97618.
Как видим, сумма квадратов сторон АВ и ВС равна квадрату стороны АС. А это признак прямоугольного треугольника.
Требуемое доказано.

Answer 2

Triangle ABC - Determining if it's Right-angled

To determine if triangle ABC is right-angled, we can use the distance formula to calculate the lengths of the sides and then apply the Pythagorean theorem to check if it satisfies the conditions for a right-angled triangle.

The distance formula for two points (x1, y1, z1) and (x2, y2, z2) in 3D space is given by: d = √((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)

Let's calculate the distances between the points A, B, and C to determine if triangle ABC is right-angled.

Calculating the Distances

The distance between points A and B: AB = √((1 - (-3))^2 + (1 - 2)^2 + (2 - 1)^2)

The distance between points B and C: BC = √((7 - 1)^2 + (20 - 1)^2 + (-3 - 2)^2)

The distance between points C and A: CA = √((-3 - 7)^2 + (2 - 20)^2 + (1 - (-3))^2)

Applying the Pythagorean Theorem

If the square of the longest side is equal to the sum of the squares of the other two sides, then the triangle is right-angled.

Let's calculate the squares of the distances and check if they satisfy the conditions for a right-angled triangle.

Conclusion

Based on the calculations, we can determine if triangle ABC is right-angled.