Question

Ребята выручайте(( Окружность, вписанная в равнобедренный треугольник ABC площадью 4, касается стороны AB в точке K. Перпендикуляр KH, опущенный на основание AC треугольника ABC, отсекает от него четырехугольник KBCH площадью 3. Найдите величину ∠ BAC (ответ записать в градусах).

Answer 1

Вместо неудобного четырёхугольника KBCH (s = 3) вычислим площадь треугольника АКН (s = 4-3 = 1), дополняющего KBCH до большого треугольника ABC (s = 4)
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Пусть основание треугольника АВС = 2а
И угол при основании Ф
АР = а
АН = а*cos Ф
КН = а*sin Ф
s(АКН) = 1/2 a^2*sin Ф*cos Ф = 1
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Теперь вычислим площадь треугольника АВС
Высота треугольника ВР 
ВР/АР = tg Ф
ВР = а*tg Ф
Основание АС = 2а
s(АВС) = 1/2*2а*а*tg Ф = а^2*tg Ф = 4
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Осталось решить систему уравнений
1/2 a^2*sin Ф*cos Ф = 1
а^2*tg Ф = 4
разделим первое на второе
1/2 sin Ф*cos Ф / tg Ф = 1/4
sin Ф*cos Ф / (sin Ф/cos Ф)  = 1/2
cos^2 Ф = 1/2
cos Ф = 1/√2
Ф = 45°

Answer 2

Problem Analysis

We are given a triangle ABC with an inscribed circle. The circle is tangent to side AB at point K. A perpendicular line KH is drawn from point K to the base AC of triangle ABC, dividing the triangle into two smaller triangles and a quadrilateral KBCH. The area of triangle ABC is 4, and the area of quadrilateral KBCH is 3. We need to find the measure of angle BAC in degrees.

Solution

To solve this problem, we can use the fact that the area of a triangle is equal to half the product of the lengths of two sides and the sine of the included angle. Let's denote the length of side AB as a, the length of side AC as b, and the measure of angle BAC as x.

We are given that the area of triangle ABC is 4, so we have the equation:

(1/2) * a * b * sin(x) = 4

We are also given that the area of quadrilateral KBCH is 3. Since the area of a quadrilateral can be calculated by subtracting the areas of two triangles from the area of the larger triangle, we have:

(1/2) * a * KH + (1/2) * KH * b * sin(x) = 3

We can simplify this equation by factoring out KH:

KH * [(1/2) * a + (1/2) * b * sin(x)] = 3

Now, let's solve these two equations simultaneously to find the value of x.

Solving the Equations

To solve the equations, we can use substitution. Let's solve the second equation for KH:

KH = 3 / [(1/2) * a + (1/2) * b * sin(x)]

Now, substitute this value of KH into the first equation:

(1/2) * a * b * sin(x) = 4

(1/2) * a * [3 / [(1/2) * a + (1/2) * b * sin(x)]] * b * sin(x) = 4

Simplify the equation:

3 * b * sin(x) = 8

b * sin(x) = 8 / 3

sin(x) = (8 / 3) / b

sin(x) = 8 / (3 * b)

Now, we can use the sine function to find the value of x:

x = arcsin(8 / (3 * b))

Final Answer

To find the measure of angle BAC in degrees, we need to substitute the value of b into the equation:

x = arcsin(8 / (3 * b))

Since we don't have the value of b, we cannot calculate the exact measure of angle BAC without additional information.