(Originais Teachy 2023) - Question Hard of Mathematics

In a topography class, a group of students needs to measure a steep slope to determine the terrain's inclination. They decide to use a trigonometric method that involves constructing a right triangle in which one of the angles is 90 degrees and the hypotenuse represents the inclination. To simplify the calculation, they choose a reference point at the top of the slope and measure the horizontal and vertical distances from that point to the base. If the horizontal distance is 15 meters and the vertical distance is 9 meters, determine whether the triangle formed is equilateral, isosceles, or scalene, considering the side lengths as real numbers. Additionally, identify if the triangle is possible in the topographic context, meaning if the provided measurements can actually form a triangle and explain why. In your explanation, use knowledge of trigonometry and plane geometry, considering the definitions of triangles and the conditions for a set of measurements to form a triangle.