Question

50 БАЛЛІВ.Основи прямокутної трапеції дорівнюють 3 см і 5 см. Знайдіть площу трапеції якщо її більша діагональ є бісектрисою прямого кута трапеції. ПОВНИЙ РОЗВ'ЯЗОК БУДЬ ЛАСКА!!!!!!!!!!!!!​

Answer 1

Відповідь:

Пояснення: Так як діагональ є бісектрисою, то вона ділить кут на два рівні кути по 45. Сума всіх кутів трикутника 180, з цього 180-90-45=45 виходить що трикутник рівнобедренний. Тоді бічна сторона, яка є і висотою, дорівнює більшій основі = 5

Answer 2

To find the area of a trapezoid, we can use the formula:

Area = (a + b) * h / 2

where: - a and b are the lengths of the parallel sides of the trapezoid - h is the height of the trapezoid

In this case, the lengths of the parallel sides of the trapezoid are given as 3 cm and 5 cm. However, we need to find the height of the trapezoid.

To find the height, we can use the fact that the larger diagonal of the trapezoid is the bisector of the right angle of the trapezoid. Let's denote the larger diagonal as d and the height as h.

From the properties of a trapezoid, we know that the height is related to the diagonals and the difference between the lengths of the parallel sides. Specifically, we have the following relationship:

h = sqrt(d^2 - ((b - a)^2) / 4)

where: - d is the length of the larger diagonal - a and b are the lengths of the parallel sides of the trapezoid

In this case, we are given that the larger diagonal is the bisector of the right angle of the trapezoid. This means that the length of the larger diagonal is equal to the hypotenuse of a right triangle with legs of length a and b. Using the Pythagorean theorem, we can find the length of the larger diagonal:

d = sqrt(a^2 + b^2)

Now we can substitute the values of a, b, and d into the formula for the height to find its value. Once we have the height, we can use the formula for the area of a trapezoid to find the area.

Let's calculate the area step by step:

Step 1: Calculate the length of the larger diagonal (d)

Using the given lengths of the parallel sides (a = 3 cm, b = 5 cm), we can calculate the length of the larger diagonal (d) using the Pythagorean theorem:

d = sqrt(a^2 + b^2) = sqrt(3^2 + 5^2) = sqrt(9 + 25) = sqrt(34) cm

Step 2: Calculate the height (h)

Using the length of the larger diagonal (d) and the lengths of the parallel sides (a = 3 cm, b = 5 cm), we can calculate the height (h) using the formula:

h = sqrt(d^2 - ((b - a)^2) / 4) = sqrt((sqrt(34))^2 - ((5 - 3)^2) / 4) = sqrt(34 - 4 / 4) = sqrt(34 - 1) = sqrt(33) cm

Step 3: Calculate the area (A)

Using the lengths of the parallel sides (a = 3 cm, b = 5 cm) and the height (h = sqrt(33) cm), we can calculate the area (A) using the formula:

A = (a + b) * h / 2 = (3 + 5) * sqrt(33) / 2 = 8 * sqrt(33) / 2 = 4 * sqrt(33) cm^2

Therefore, the area of the trapezoid is 4 * sqrt(33) cm^2.

Please note that the calculations provided above are based on the given information and formulas for finding the area of a trapezoid.