Question

через вершины квадрата проведены прямые, параллельные его диагоналям, которые, пересекаясь, образуют четырёхугольник. Устаеовите вид и найдите периметр четырёхугольника, если диагональ квадрата рввна 4,5 см.

Answer 1

Четырёхугольник - квадрат, со стороной 4,5 см (равна диагонали квадрата).

Соответственно периметр равен 18 см ( P=4,5 * 4)

Answer 2

Параллельные прямые не пересекаются. Как можно провести параллельную прямую диагонали через вершину квадрата, которая принадлежит диагонали? Условие не корректно.

Answer 3

Problem Analysis

We are given a square, and through its vertices, lines are drawn parallel to its diagonals, forming a quadrilateral. We need to determine the shape of the quadrilateral and find its perimeter, given that the diagonal of the square is 4.5 cm.

Quadrilateral Shape

To determine the shape of the quadrilateral formed by the lines through the vertices of the square, we need to analyze the properties of the lines and the square.

From the given information, we know that the lines are parallel to the diagonals of the square. This means that the lines divide the square into two smaller congruent squares, as shown in the diagram below:

   A _______ B
    |       |
    |   Q   |
    |_______|
   D         C

In the diagram, A, B, C, and D represent the vertices of the square, and Q represents the quadrilateral formed by the lines through the vertices.

Since the lines divide the square into two congruent squares, the quadrilateral Q is also a square. Therefore, the shape of the quadrilateral is a square.

Finding the Perimeter of the Quadrilateral

To find the perimeter of the quadrilateral, we need to determine the length of its sides. Since the quadrilateral is a square, all four sides are equal in length.

To find the length of the sides, we can use the information given about the diagonal of the square. The diagonal of a square divides it into two congruent right triangles. We can use the Pythagorean theorem to find the length of the sides.

Let's denote the length of the sides of the square as s. The diagonal of the square is given as 4.5 cm. Using the Pythagorean theorem, we have:

s^2 + s^2 = (4.5)^2

Simplifying the equation:

2s^2 = 20.25

Dividing both sides by 2:

s^2 = 10.125

Taking the square root of both sides:

s = √10.125 ≈ 3.18 cm

Therefore, the length of each side of the quadrilateral is approximately 3.18 cm.

Since the quadrilateral is a square, all four sides are equal in length. Therefore, the perimeter of the quadrilateral is:

Perimeter = 4 * s = 4 * 3.18 cm = 12.72 cm

Answer

The quadrilateral formed by the lines through the vertices of the square is a square. The perimeter of the quadrilateral is approximately 12.72 cm.

Please let me know if you need any further clarification.