Question

Один из внешних углов прямоугольного треугольника равен 120.Сумма длин гипотенузы и меньшего катета треугольника равна 18см.Найдите длину каждой стороны треугольника. 2) Треугольник ABC прямоугольный,ВС-гипотенуза, АD-высота, уголВ=60,DB=2см. Найдите длину отрезка DC. 3)Один из острых углов прямоугольного треугольника равен 60. Сумма малого катета и гипотенузы равна 2,64см. Найдите Доину гипотенузы треугольника. ПОЖАЛУЙСТА РЕШИТЕ ХОТЯБ 2 ЗАРАНЕЕ СПАСИБО ЭТО ДЛЯ МЕНЯ ОЧЕНЬ ВАЖНО РЕШИТЕЕЕ ПРОШУУУ

Answer 1

У Вас все задачи на на прямоугольный треугольник с углами 90, 60 и 30  градусов.
Нарисуйте себе, чтоб понять задачи, такой треугольник, где углы 90, 60, 30 градусов. Этот треугольник ровно половина равностороннего треугольника, у которого все углы по 60 градусов. Сразу поймёте.
1).    Итак, рассмотрим прямоугольный треугольник , внешний угол равен 120 градусов, значит смежный угол в треугольнике равен 60 градусов. Один угол в треугольнике равен 90 град. по условию, значит третий угол равен 180 - 90 - 60 = 30 (градусов). Гипотенуза в таком треугольнике равна: малый катет а умножить на два: а х 2,  с  - это гипотенуза, b - это большой катет,
в условии сказано, что а + с =18,  в треугольнике с такими углами с / а = 2 / 1
Отсюда а = 18 / 3 = 6 (см) ; с = 18 - 6 = 12 (см); b² = c² - a² = 12² - 6² = 144 - 36 = 108.
b = √108 = √36·3 = 6√3   
2).     АВ = 2·ВD = 2 · 2 = 4 (см);   ВС = 2 АВ = 4 · 2 = 8 (см);
DC = 8 - 2 = 6 (см)
3).     Рассматриваем всё тот же прямоугольный треугольник с углами 90, 60 и 30 градусов,  где    a    и    b  -   катеты,  а   с   -  гипотенуза.
a + с =2,64 см     по условию
a : с = 1 : 2  (малый катет относится к гипотенузе как один к двум)
т.е. ( а + с )   это три части,   а   с - гипотенуза составляет  2 части
 с = ( 2,64 /3) · 2 = 0,88 · 2 = 1,76 (см)

Answer 2

Task 1: Finding the lengths of the sides of a right triangle

To find the lengths of the sides of a right triangle, we are given the following information:

- One of the external angles of the right triangle is 120 degrees. - The sum of the hypotenuse and the smaller leg of the triangle is 18 cm.

Let's denote the sides of the triangle as follows: - The hypotenuse: c - The smaller leg: a - The larger leg: b

To solve this problem, we can use the trigonometric ratios for a right triangle. Specifically, we can use the sine, cosine, and tangent ratios.

Using the given information, we can determine the following:

1. The angle opposite the smaller leg is 90 - 120 = 30 degrees. 2. The angle opposite the larger leg is 180 - 90 - 30 = 60 degrees.

Now, let's solve for the lengths of the sides of the triangle:

1. The length of the smaller leg, a, can be found using the sine ratio: - sin(30) = a / c - a = c * sin(30)

2. The length of the larger leg, b, can be found using the cosine ratio: - cos(30) = b / c - b = c * cos(30)

3. The sum of the hypotenuse and the smaller leg is given as 18 cm: - c + a = 18

Now, let's substitute the values and solve for the lengths of the sides of the triangle.

Solution:

1. From the equation c + a = 18, we can solve for c: - c = 18 - a

2. Substituting the value of c in terms of a into the equation b = c * cos(30), we get: - b = (18 - a) * cos(30)

3. Substituting the value of c in terms of a into the equation a = c * sin(30), we get: - a = (18 - a) * sin(30)

Simplifying the equations, we have: - b = (18 - a) * cos(30) - a = (18 - a) * sin(30)

Now, we can solve these equations simultaneously to find the values of a and b.

Please note that the exact values of a and b cannot be determined without additional information. However, we can solve for the values in terms of a.

Let's solve the equations:

b = (18 - a) * cos(30) a = (18 - a) * sin(30)

Simplifying the equations further, we have:

b = 18cos(30) - acos(30) a = 18sin(30) - asin(30)

Solving for a and b:

a = 18sin(30) - asin(30) a + asin(30) = 18sin(30) a(1 + sin(30)) = 18sin(30) a = 18sin(30) / (1 + sin(30))

b = 18cos(30) - acos(30) b = 18cos(30) - a * cos(30) b = 18cos(30) - (18sin(30) / (1 + sin(30))) * cos(30)

Therefore, the lengths of the sides of the triangle are:

- The hypotenuse, c = 18 - a - The smaller leg, a = 18sin(30) / (1 + sin(30)) - The larger leg, b = 18cos(30) - (18sin(30) / (1 + sin(30))) * cos(30)

Please note that these values are in terms of a and cannot be simplified further without additional information.

Task 2: Finding the length of segment DC

In this task, we are given the following information about triangle ABC:

- Triangle ABC is a right triangle. - BC is the hypotenuse. - AD is the height. - Angle B is 60 degrees. - DB = 2 cm.

We need to find the length of segment DC.

To solve this problem, we can use trigonometric ratios for a right triangle.

Let's denote the length of segment DC as x.

Using the given information, we can determine the following:

1. The angle opposite segment DC is 180 - 90 - 60 = 30 degrees.

Now, let's solve for the length of segment DC:

1. The length of segment DC, x, can be found using the tangent ratio: - tan(30) = x / 2 - x = 2 * tan(30)

Now, let's substitute the value and solve for the length of segment DC.

Solution:

1. Substituting the value of x in terms of tan(30), we get: - x = 2 * tan(30)

Therefore, the length of segment DC is x = 2 * tan(30).

Please note that the exact value of x cannot be determined without additional information. However, we can calculate the approximate value using a calculator or trigonometric tables.

Task 3: Finding the length of the hypotenuse of a right triangle

In this task, we are given the following information about a right triangle:

- One of the acute angles is 60 degrees. - The sum of the smaller leg and the hypotenuse is 2.64 cm.

We need to find the length of the hypotenuse of the triangle.

To solve this problem, we can use trigonometric ratios for a right triangle.

Let's denote the length of the hypotenuse as c.

Using the given information, we can determine the following:

1. The angle opposite the smaller leg is 90 - 60 = 30 degrees.

Now, let's solve for the length of the hypotenuse:

1. The length of the hypotenuse, c, can be found using the cosine ratio: - cos(30) = c / (a + c) - c = (a + c) * cos(30)

2. The sum of the smaller leg and the hypotenuse is given as 2.64 cm: - a + c = 2.64

Now, let's substitute the values and solve for the length of the hypotenuse.

Solution:

1. From the equation a + c = 2.64, we can solve for a: - a = 2.64 - c

2. Substituting the value of a in terms of c into the equation c = (a + c) * cos(30), we get: - c = (2.64 - c + c) * cos(30) - c = 2.64 * cos(30)

Therefore, the length of the hypotenuse is c = 2.64 * cos(30).

Please note that the exact value of c cannot be determined without additional information. However, we can calculate the approximate value using a calculator or trigonometric tables.