Question

Дан правильный 16-угольник. Найдите количество четвёрок его вершин, являющихся вершинами выпуклого четырёхугольника, в котором ровно два угла равны 90. (Две четвёрки вершин, отличающиеся порядком вершин, считаются одинаковыми.)

Answer 1

Трапецию с двумя вершинами по 90 градусов не вписать в правильный 16-и угольник. зато можно вписать четырёхугольника, у которого два противоположных угла по 90°
Рассмотрим диагональ такого четырёхугольника, проведённую из вершины тупого угла в острый угол. Квадраты под запретом по условию. Эта диагональ является одновременно диаметром описанной окружности 16-и угольника и четырёхугольника, и гипотенузой двух прямоугольных треугольников, на которые диагональ делит четырёхугольник.
Всего диагоналей возможно 16/2=8
С каждой стороны от диагонали возможны 7 точек расположения прямого угла.
И всего четырёхугольников возможно
7*7*8=49*8=392

Answer 2

Calculation of the Number of Vertices in a Convex Quadrilateral with Two Right Angles

To find the number of vertices in a convex quadrilateral with exactly two right angles, we need to consider the properties of a 16-gon and the conditions for a convex quadrilateral.

A 16-gon is a polygon with 16 sides and 16 vertices. In a convex quadrilateral, the sum of the interior angles is always 360 degrees. Since we are looking for a quadrilateral with exactly two right angles (90 degrees each), the sum of the other two angles must be 180 degrees.

To determine the number of vertices that can form a convex quadrilateral with two right angles in a 16-gon, we need to find the number of ways we can choose two vertices to form the right angles.

Let's calculate the number of ways to choose two vertices from a 16-gon:

Step 1: Calculate the total number of ways to choose two vertices from a 16-gon. The total number of ways to choose two vertices from a 16-gon is given by the combination formula: C(16, 2) = 16! / (2! * (16-2)!) = 16! / (2! * 14!) = (16 * 15) / (2 * 1) = 120.

Step 2: Calculate the number of ways to choose two vertices to form a right angle. Since we need to choose two vertices to form a right angle, the number of ways to choose two vertices is given by the combination formula: C(2, 2) = 2! / (2! * (2-2)!) = 2! / (2! * 0!) = 1.

Step 3: Calculate the number of ways to choose the remaining two vertices. To form a convex quadrilateral, we need to choose the remaining two vertices from the remaining 14 vertices (excluding the two vertices already chosen for the right angles). The number of ways to choose two vertices from 14 is given by the combination formula: C(14, 2) = 14! / (2! * (14-2)!) = 14! / (2! * 12!) = (14 * 13) / (2 * 1) = 91.

Step 4: Calculate the total number of vertices that can form a convex quadrilateral with two right angles. The total number of vertices that can form a convex quadrilateral with two right angles in a 16-gon is given by the product of the number of ways to choose two vertices to form a right angle and the number of ways to choose the remaining two vertices: 1 * 91 = 91.

Therefore, the number of vertices in a 16-gon that can form a convex quadrilateral with exactly two right angles is 91.