Question

Прямоугольник разбили на 4 прямоугольные клетки вертикальной и горизонтальной прямыми.У трёх клеток периметры равны 3,4,5.Найдите периметр четвёртой клетки, если известно, что из всех клеток она имеет самый большой периметр.

Answer 1

Работаем через полупериметр

Answer 2

Problem Analysis

We are given a rectangle that has been divided into 4 rectangular cells by vertical and horizontal lines. The perimeters of three of these cells are given as 3, 4, and 5. We need to find the perimeter of the fourth cell, knowing that it has the largest perimeter among all the cells.

Solution

Let's assume the lengths of the sides of the rectangle are a and b. We can represent the perimeters of the four cells as follows:

Cell 1: 2a + 2x Cell 2: 2b + 2y Cell 3: 2a + 2z Cell 4: 2b + 2w

We are given the perimeters of cells 1, 2, and 3 as 3, 4, and 5, respectively. Let's denote them as p1, p2, and p3.

p1 = 2a + 2x = 3 p2 = 2b + 2y = 4 p3 = 2a + 2z = 5

We need to find the perimeter of cell 4, which is represented by p4.

To find p4, we can rearrange the equations for p1 and p3 to solve for a and z:

a = (p1 - 2x) / 2 z = (p3 - 2a) / 2

Substituting the values of a and z into the equation for p4, we get:

p4 = 2b + 2w = 2b + 2(p3 - 2a) / 2 = 2b + p3 - 2a

Now, we need to find the maximum value of p4. To do this, we can substitute the values of a and z into the equation for p4 and simplify:

p4 = 2b + p3 - 2a = 2b + p3 - 2((p1 - 2x) / 2) = 2b + p3 - (p1 - 2x) = 2b + p3 - p1 + 2x

Since we want to find the maximum value of p4, we need to maximize the expression 2b + p3 - p1 + 2x. To do this, we can maximize each term individually.

Since b, p3, and p1 are given constants, we need to maximize 2x. The maximum value of 2x occurs when x is the largest possible value. Since the rectangle is divided into four cells, x cannot be greater than half the length of the rectangle. Therefore, the maximum value of 2x is equal to the length of the rectangle.

Finally, substituting the values into the equation for p4, we get:

p4 = 2b + p3 - p1 + 2x = 2b + p3 - p1 + 2 * (length of the rectangle)

Therefore, the perimeter of the fourth cell is equal to 2b + p3 - p1 + 2 times the length of the rectangle.

Answer

The perimeter of the fourth cell, knowing that it has the largest perimeter among all the cells, is equal to 2b + p3 - p1 + 2 times the length of the rectangle.

Please note that we do not have the specific values of b, p3, and p1, so we cannot provide a numerical answer. However, you can substitute the given values into the equation to calculate the perimeter of the fourth cell.