Question

3. У прямокутному трикутнику катет завдовжки 10 см є протилежним до кута 30°. Знайдіть гіпотенузу трикутника. 4. У прямокутному трикутнику гіпотенуза дорівнює 30 см, а синус одного з кутів — 0,6. Знайдіть катети трикутника Срочно

Answer 1

Ответ:

Відповідь: номер 3: АВ=5

номер 4: АС=18, СВ=24

Answer 2

Finding the Hypotenuse of a Right Triangle

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

In this case, we are given that one of the legs (or catheti) of the right triangle has a length of 10 cm and is opposite a 30° angle. Let's denote this leg as a and the hypotenuse as c.

Using the Pythagorean theorem, we can set up the following equation:

a^2 + b^2 = c^2

Since we know the length of leg a (10 cm) and the angle opposite it (30°), we can use trigonometric ratios to find the length of the other leg (b) and then calculate the hypotenuse (c).

Finding the Other Leg of the Triangle

To find the length of the other leg (b), we can use the sine function, which relates the ratio of the length of the side opposite an angle to the length of the hypotenuse. The sine of an angle is equal to the length of the side opposite the angle divided by the length of the hypotenuse.

In this case, we are given that the sine of the angle opposite leg a is 0.6. Let's denote the length of leg b as x.

Using the sine function, we can set up the following equation:

sin(30°) = x / c

Substituting the given values, we have:

0.6 = x / c

Solving for x, we get:

x = 0.6c

Solving for the Hypotenuse

Now that we have expressions for both legs of the triangle, we can substitute them into the Pythagorean theorem equation and solve for the hypotenuse (c).

Substituting the values of a and b into the Pythagorean theorem equation, we have:

(10)^2 + (0.6c)^2 = c^2

Simplifying the equation, we get:

100 + 0.36c^2 = c^2

Rearranging the equation, we have:

0.36c^2 - c^2 = -100

-0.64c^2 = -100

Dividing both sides of the equation by -0.64, we get:

c^2 = 156.25

Taking the square root of both sides, we find:

c = 12.5

Therefore, the length of the hypotenuse of the triangle is 12.5 cm.

Finding the Other Leg of the Triangle

To find the length of the other leg (b), we can substitute the value of c into the expression we found earlier:

x = 0.6c

Substituting c = 12.5, we have:

x = 0.6 * 12.5

Simplifying the expression, we find:

x = 7.5

Therefore, the length of the other leg of the triangle is 7.5 cm.

In summary, for the given right triangle with a leg of 10 cm opposite a 30° angle, the length of the hypotenuse is 12.5 cm, and the length of the other leg is 7.5 cm.