Question

Знайдіть площу прямокутника якщо сума двох його непаралел них сторін дорівнєю 14 см а діагональ дорівнює 10 см пжжж поможіть дам 20 балів))​

Answer 1

Дано:

ABCD — прямокутник;

AB + AD = 14 см;

BD = 10 см.

Знайти:

S

                                              Розв'язок:

AB + AD = 14.

Піднесемо обидві частини цієї рівності до квадрата:

(AB + AD)² = 14²

Розкриємо дужки за формулою квадрата суми:

AB² + 2 · AB · AD + AD² = 196.

   Запишемо формулу площі прямокутника АВСD:

   S = AB · AD.

   Δ BAD — прямокутний (бо ∠BAD = 90°).

   За теоремою Піфагора:

   AB² + AD² = BD² = 10² = 100.

Таким чином, AB · AD = S; AB² + AD² = 100. Підставимо ці значення у рівняння:

100 + 2 · S = 196

2 · S = 196 - 100

2 · S = 96

S = 96 : 2

S = 48 (см²).

Відповідь: 48 см.

Answer 2

Finding the Area of a Rectangle

To find the area of a rectangle, we need to know the lengths of its sides. In this case, we are given that the sum of two non-parallel sides of the rectangle is 14 cm, and the diagonal is 10 cm.

Let's assume the lengths of the two non-parallel sides are a and b. The diagonal of the rectangle forms a right triangle with the sides a, b, and the diagonal itself.

Using the Pythagorean theorem, we can express the relationship between the sides of the right triangle:

a^2 + b^2 = diagonal^2

In this case, the diagonal is given as 10 cm, so we have:

a^2 + b^2 = 10^2

Simplifying further:

a^2 + b^2 = 100 ---(1)

We are also given that the sum of the two non-parallel sides is 14 cm:

a + b = 14 ---(2)

To find the area of the rectangle, we need to multiply the lengths of the two non-parallel sides:

Area = a * b

Now, let's solve the system of equations (1) and (2) to find the values of a and b.

Solving the System of Equations

We can solve the system of equations (1) and (2) using substitution or elimination method. Let's use the substitution method.

From equation (2), we can express a in terms of b:

a = 14 - b

Substituting this value of a into equation (1):

(14 - b)^2 + b^2 = 100

Expanding and simplifying:

196 - 28b + b^2 + b^2 = 100

Combining like terms:

2b^2 - 28b + 96 = 0

Dividing the equation by 2 to simplify:

b^2 - 14b + 48 = 0

Now, we can solve this quadratic equation to find the values of b.

Using the quadratic formula:

b = (-(-14) ± √((-14)^2 - 4 * 1 * 48)) / (2 * 1)

Simplifying further:

b = (14 ± √(196 - 192)) / 2

b = (14 ± √4) / 2

b = (14 ± 2) / 2

This gives us two possible values for b:

b = (14 + 2) / 2 = 8

b = (14 - 2) / 2 = 6

Now, substituting these values of b back into equation (2) to find the corresponding values of a:

For b = 8:

a = 14 - 8 = 6

For b = 6:

a = 14 - 6 = 8

So, we have two possible sets of values for the lengths of the non-parallel sides of the rectangle:

1. a = 6 cm and b = 8 cm 2. a = 8 cm and b = 6 cm

Calculating the Area

Now that we have the lengths of the non-parallel sides of the rectangle, we can calculate the area using the formula:

Area = a * b

For the first set of values (a = 6 cm and b = 8 cm):

Area = 6 cm * 8 cm = 48 cm^2

For the second set of values (a = 8 cm and b = 6 cm):

Area = 8 cm * 6 cm = 48 cm^2

Therefore, the area of the rectangle is 48 square centimeters for both sets of values.

I hope this helps! Let me know if you have any further questions.