Question

Площадь прямоугольника ABCD равна 48 см2. Все стороны прямоугольника увеличили в 1.5 раза. После изменений сторон квадрат диагонали прямоугольника увеличилась на 125 см2. Каким был исходный периметр прямоугольника?

Answer 1

Ответ:   P=28 см²

Пусть длина прямоугольника равна х см , а ширина равна у см .

Диагональ прямоугольника d  , по теореме Пифагора   $d^2=x^2+y^2$  .

Площадь прямоугольника равна  $S=xy$   ⇒   $xy=48\ \ \to \ \ \ y=\dfrac{48}{x}$  .

Стороны нового прямоугольника равны  1,5х см и  1,5у см .

Квадрат длины диагонали нового прямоугольника равен  

$d_1^2=(1,5x)^2+(1,5y)^2\ \ ,\ \ \ d_1^2=2,25x^2+2,25y^2\ \ ,\ \ d_1^2=2,25\, (x^2+y^2)$

По условию  $d_1^2-d^2=125$  .

$2,25\, (x^2+y^2)-(x^2+y^2)=125\\\\1,25(x^2+y^2)=125\\\\x^2+y^2=100$

$x^2+\dfrac{48^2}{x^2}=100\ \ \to \ \ \ \dfrac{x^4-100x^2+2304}{x^2}=0\ \ \ (x\ne 0)\ \ ,\\\\\\x^4-100x^2+2304=0\ \ ,\\\\t=x^2\geq 0\ \ ,\ \ t^2+100t-2304=0\ \ ,\ \ D/4=50^2+2304=196=14^2\ ,\\\\t_1=50-14=36\ \ ,\ \ t_2=50+14=64\\\\x^2=36\ \ \ \to \ \ \ x_{1,2}=\pm 6\\\\x^2=64\ \ \ \to \ \ \ x_{3,4}=\pm 8$

Отрицательные значения переменная не может принимать из-за геометрического смысла переменной .

$x=6\ \ \to \ \ \ y=\dfrac{48}{6}=8\\\\x=8\ \ \ \to \ \ \ y=\dfrac{48}{6}=6$  

Стороны прямоугольника были равны 8 см и 6 см .

Периметр первого прямоугольника равен

$P=2\cdot (8+6)=2\cdot 14=\bf 28$  см .

Answer 2

Calculation of the Original Rectangle's Perimeter

To find the original perimeter of the rectangle, we need to determine the lengths of its sides.

Let's assume the original length of the rectangle is x cm and the original width is y cm.

According to the given information, the area of the rectangle is 48 cm². Therefore, we have the equation:

x * y = 48 After increasing the lengths of all sides by 1.5 times, the new length of the rectangle becomes 1.5x cm and the new width becomes 1.5y cm.

We are also given that the square of the diagonal of the rectangle increased by 125 cm². The square of the diagonal can be calculated using the Pythagorean theorem:

Diagonal² = Length² + Width²

The original square of the diagonal is x² + y² cm², and after the increase, it becomes (1.5x)² + (1.5y)² cm². The difference between these two values is 125 cm².

So, we have the equation:

(1.5x)² + (1.5y)² - (x² + y²) = 125 To find the original perimeter, we need to calculate the sum of all four sides of the rectangle:

Perimeter = 2 * (Length + Width)

Let's solve the equations to find the values of x and y, and then calculate the original perimeter.

Solving the Equations

We can simplify the equations and solve them simultaneously to find the values of x and y.

From equation 1, we have:

x * y = 48

From equation 2, we have:

(1.5x)² + (1.5y)² - (x² + y²) = 125

Expanding the equation, we get:

2.25x² + 2.25y² - x² - y² = 125

Combining like terms, we have:

1.25x² + 1.25y² = 125

Dividing both sides by 1.25, we get:

x² + y² = 100

Now, we have a system of equations:

x * y = 48 x² + y² = 100

We can solve this system of equations to find the values of x and y.

Solution

Using the first equation, we can express y in terms of x:

y = 48 / x

Substituting this into the second equation, we have:

x² + (48 / x)² = 100

Simplifying the equation, we get:

x⁴ - 100x² + 2304 = 0

This is a quadratic equation in terms of x². We can solve it using the quadratic formula:

x² = (-b ± √(b² - 4ac)) / (2a)

For our equation, a = 1, b = -100, and c = 2304.

Solving the quadratic equation, we find two possible values for x²:

x² = 64 or x² = 36

Taking the square root of both sides, we get:

x = 8 or x = 6

Substituting these values back into the equation y = 48 / x, we find:

For x = 8, y = 48 / 8 = 6 For x = 6, y = 48 / 6 = 8

Therefore, the possible dimensions of the original rectangle are:

1. Length = 8 cm, Width = 6 cm 2. Length = 6 cm, Width = 8 cm

Calculating the Original Perimeter

Now that we have the possible dimensions of the original rectangle, we can calculate the original perimeter using the formula:

Perimeter = 2 * (Length + Width)

For the first set of dimensions (Length = 8 cm, Width = 6 cm):

Perimeter = 2 * (8 + 6) = 2 * 14 = 28 cm

For the second set of dimensions (Length = 6 cm, Width = 8 cm):

Perimeter = 2 * (6 + 8) = 2 * 14 = 28 cm

Therefore, the original perimeter of the rectangle was 28 cm.

Please note that there may be other valid solutions depending on the specific values of x and y.