Question

ПОМОГИТЕ ПОЖАЛУЙСТА СРОЧНО ДАЮ 50 БАЛЛОВ!!!!! АЛГЕБРА 8 КЛАСС Діагональ прямокутника на 8 см більша за одну з його сторін і на 4 см більша за другу. Знайдіть площу прямокутника. У відповідь запишіть числове значення площі.​

Answer 1

Ответ:

Площа прямокутника дорівнює 260 квадратних сантиметрів.

Объяснение:

Нехай сторони прямокутника дорівнюють x та y, де x - менша сторона. Тоді за теоремою Піфагора маємо:

d^2 = x^2 + y^2,

де d - діагональ.

Також маємо систему рівнянь:

d = x + 8,

d = y + 4.

Підставляємо друге рівняння у перше:

(y+4)^2 + x^2 = (x+8)^2,

y^2 + 8y + 16 + x^2 = x^2 + 16x + 64,

8y = 16x + 48,

y = 2x + 6.

Підставляємо знайдене значення y у другу систему рівнянь:

d = y + 4,

x + 8 = y + 4 + 8,

x + 8 = y + 12,

x = y - 4.

Отже, маємо систему:

y = 2x + 6,

x = y - 4.

Розв'язавши її, знаходимо x = 10 і y = 26.

Площа прямокутника S = xy = 10 * 26 = 260.

Отже, площа прямокутника дорівнює 260 квадратних сантиметрів.

Answer 2

Finding the Length of the Rectangle's Diagonals

To find the area of the rectangle, we first need to determine the lengths of its sides. The problem states that one diagonal of the rectangle is 8 cm longer than one of its sides, and the other diagonal is 4 cm longer than the other side.

Let's assume that the shorter side of the rectangle has a length of x cm. Therefore, the longer side will be x + 8 cm, and the other diagonal will be x + 4 cm longer than the shorter side.

Applying the Pythagorean Theorem

To find the lengths of the diagonals, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In this case, the diagonals of the rectangle form right triangles with the sides of the rectangle. Let's label the shorter side as a, the longer side as b, and the diagonals as d1 and d2.

For the first right triangle: - One side is a cm. - The other side is b cm. - The hypotenuse is d1 cm, which is equal to x + 8 cm.

Using the Pythagorean theorem, we can write the equation: a^2 + b^2 = d1^2.

For the second right triangle: - One side is b cm. - The other side is a cm. - The hypotenuse is d2 cm, which is equal to x + 4 cm.

Using the Pythagorean theorem, we can write the equation: b^2 + a^2 = d2^2.

Solving the Equations

To find the lengths of the diagonals, we need to solve the system of equations formed by the two Pythagorean theorem equations.

Let's solve the first equation for b^2: b^2 = d1^2 - a^2.

Substituting this expression for b^2 in the second equation, we get: (d1^2 - a^2) + a^2 = d2^2.

Simplifying the equation, we have: d1^2 - a^2 + a^2 = d2^2.

The a^2 terms cancel out, leaving us with: d1^2 = d2^2.

Since the squares of the diagonals are equal, the diagonals must have the same length.

Finding the Area of the Rectangle

Now that we know the diagonals have the same length, we can find the area of the rectangle using the formula: Area = (1/2) * d1 * d2.

Since the diagonals have the same length, we can simplify the formula to: Area = (1/2) * d^2.

Substituting the value of d1 or d2 (since they are equal) with x + 8 or x + 4, we get: Area = (1/2) * (x + 8)^2.

Simplifying further, we have: Area = (1/2) * (x^2 + 16x + 64).

To find the numerical value of the area, we need to know the value of x. Unfortunately, the problem does not provide this information. Therefore, we cannot determine the exact numerical value of the area of the rectangle without knowing the value of x.

Please provide the value of x so that we can calculate the area of the rectangle.