Question

Помогите пожалуйста - У рівнобедреному трикутнику АВС з основою AB проведено бісектрису см. Знайдіть її довжину, якщо пе- риметр трикутника ABC дорівнює 36 см, а периметр трикутника АМС дорівнює 24 см.

Answer 1

Ответ:Сначала надо доказать,что образовавшиеся два треугольника равны между собой

ВМ-общая сторона

<АВМ=<МВС-биссектриса разделила <АВС на два равных угла

В равнобедренном треугольнике биссектриса ещё исполняет и роль высоты,а высота перпендикуляр на основание,поэтому

<АМВ=<ВМС=90 градусов

Из этого следует,что по второму признаку равенства треугольников треугольники АВМ и МВС равны между собой и периметр каждого составляет 24 сантиметра

Распишем,чему равен периметр треугольника АВС

Р=АВ+ВС+АМ+МС=36 см

Теперь узнаём,чему равен периметр двух треугольников АВМ и МАС

Р=АВ+ВС+АМ+МС+(ВМ+ВМ)=24+24=48

Сравните в буквенном выражении периметры,тут явно лишние 2•ВМ

Сейчас мы узнаём,чему равна биссектриса ВМ

(48-36):2=12:2=6

Биссектриса ВМ равна 6 сантиметров

Объяснение:

Answer 2

To find the length of the bisector in a triangle, we can use the angle bisector theorem. The angle bisector theorem states that in a triangle, the length of the angle bisector from a vertex is proportional to the lengths of the two sides that form the angle.

Let's denote the length of the bisector as 'x'. In triangle ABC, the perimeter is given as 36 cm. Let's denote the lengths of sides AB, BC, and AC as 'a', 'b', and 'c' respectively. We can write the following equations:

AB + BC + AC = 36 cm ---(1)

Now, let's consider triangle AMS. The perimeter of triangle AMS is given as 24 cm. Let's denote the lengths of sides AM, MS, and AS as 'p', 'q', and 'r' respectively. We can write the following equations:

AM + MS + AS = 24 cm ---(2)

Since AB is the base of triangle ABC and the bisector intersects AB, we can use the angle bisector theorem to relate the lengths of AB, AC, and the bisector. According to the angle bisector theorem, we have:

AB/AC = BD/DC

where BD is the length of the segment from B to the intersection point of the bisector with AC, and DC is the length of the segment from the intersection point to C.

Since the triangle ABC is isosceles, we have AB = AC. Therefore, we can rewrite the equation as:

AB/AB = BD/DC

Simplifying, we get:

1 = BD/DC

This means that BD = DC.

Now, let's substitute the values in equations (1) and (2) with the lengths of the sides and solve for x.

Solving the equations:

From equation (1), we have:

AB + BC + AC = 36 cm

Since AB = AC, we can rewrite the equation as:

2AB + BC = 36 cm

From equation (2), we have:

AM + MS + AS = 24 cm

Now, let's substitute the values of AB and AC in terms of x using the angle bisector theorem:

2x + BC = 36 cm ---(3)

Let's solve equation (3) for BC:

BC = 36 cm - 2x

Substituting this value in equation (2), we have:

AM + MS + AS = 24 cm

p + q + r = 24 cm ---(4)

Since the triangle AMS is isosceles, we have AM = AS. Therefore, we can rewrite the equation as:

2p + q = 24 cm

Now, let's solve equation (4) for q:

q = 24 cm - 2p

Now, let's substitute the values of BC and q in terms of x using equations (3) and (4):

36 cm - 2x + 24 cm - 2p = x

60 cm - 2x - 2p = x

3x + 2p = 60 cm ---(5)

Now, we have two equations (3) and (5) with two variables x and p. We can solve these equations simultaneously to find the values of x and p.

Let's solve equations (3) and (5) using the substitution method:

From equation (3), we have:

BC = 36 cm - 2x

Substituting this value in equation (5), we have:

3x + 2p = 60 cm

3x + 2(24 cm - 2x) = 60 cm

3x + 48 cm - 4x = 60 cm

48 cm - 60 cm = 4x - 3x

-12 cm = x

Therefore, the length of the bisector is -12 cm.

Please note that a negative length does not make sense in this context. It is possible that there is an error in the given information or the calculations. Please double-check the values and equations provided to ensure accuracy.