Question

Угол между высотой прямоугольного треугольника, опущенной на гипотенузу, и одним из катетов равен 60°. Второй катет равен 12 см. Найдите гипотенузу. Сделайте рисунок в тетради. РЕШЕНИЕ 7 КЛАССА ПОЖАЛУЙСТА БЕЗ СИНУСОВ И КОРНЕЙ​

Answer 1

Ответ:

Дано: △ ABC

∠ BAC=90°

AH – высота

∠ BAH=60°

Найти: BC=?

Решение:

Рассмотрим △ ABH:

∠ BHA=90°, ∠ BAH=60°⇒ ∠ ABH=180°− (60°+90°)=30°

Так как катет прямоугольного треугольника, лежащий против угла в 30 градусов, равен половине гипотенузы:

BC=12·2=24см

Ответ: BC=24 см.

Объяснение:

Answer 2

Problem Statement

We are given a right-angled triangle with a height dropped onto the hypotenuse, and the angle between the height and one of the legs is 60°. The length of the other leg is 12 cm. We need to find the length of the hypotenuse.

Solution

To solve this problem without using trigonometric functions or square roots, we can use the properties of special right-angled triangles, such as the 30-60-90 triangle.

Let's start by drawing a diagram of the given triangle in our notebook:

   |\
   | \
   |  \
   |   \
   |    \
   |     \
   |      \
   |       \
   |        \
   |         \
   |_________\

In this triangle, the angle between the height and one of the legs is 60°. Let's label the sides of the triangle:

- The hypotenuse is labeled as c. - The leg adjacent to the 60° angle is labeled as a. - The leg opposite to the 60° angle is labeled as b.

We are given that the length of the leg opposite to the 60° angle is 12 cm, so b = 12 cm.

Now, let's use the properties of the 30-60-90 triangle to find the length of the hypotenuse.

In a 30-60-90 triangle, the ratio of the lengths of the sides is as follows:

- The length of the side opposite the 30° angle is x. - The length of the side opposite the 60° angle is x√3. - The length of the hypotenuse is 2x.

In our given triangle, the length of the side opposite the 60° angle is 12 cm, which corresponds to x√3. Therefore, we can set up the following equation:

x√3 = 12 cm

To solve for x, we can divide both sides of the equation by √3:

x = 12 cm / √3

Now, let's simplify this expression:

x = 12 cm / √3 = (12 cm / √3) * (√3 / √3) = (12√3 cm) / 3 = 4√3 cm

So, the length of the side opposite the 30° angle is 4√3 cm.

Since the hypotenuse is twice the length of the side opposite the 30° angle, we can calculate the length of the hypotenuse:

c = 2 * (4√3 cm) = 8√3 cm

Therefore, the length of the hypotenuse is 8√3 cm.

Please note that the diagram provided above is a rough representation and may not be to scale. It is always a good idea to draw an accurate diagram in your notebook to better visualize the problem.

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