Question

Задание 1. НУЖЕН РИСУНОКДве стороны треугольника равны 3 см и 6 см, а угол между ними составляет 60°. Определите:а) длину третьей стороны треугольника (10 баллов);б) периметр треугольника (10 баллов);в) площадь треугольника (10 баллов);г) радиус окружности, описанной около треугольника (10 баллов).ЛЮДИ ДОБРЫЕ МНЕ НЕ НУЖНО РЕШЕНИЕ МНЕ НУЖЕН РИСУНОК

Answer 1

а) Опустим высоту АН из вершины угла, и рассмотрим получившийся прямоугольный треугольник АВН,

{< - угол}

<Н=90°, по определению прямоугольного треугольника, зная сумму всех углов этого треугольника, найдем <ВАН

<ВАН=90°-60°=30°

Против угла в 30° лежит катет равный половине гипотенузы, а значит ВН=0,5*3=1,5

Найдем АН по теореме Пифагора

$AH = \sqrt{ AB {}^{2} - BH {}^{2} } = \\ = \sqrt{9 - 2.25} = \sqrt{6.75} ≈2.6$

Найдем НС, зная ВН и ВС,

$HC = 6 - 1.5 = 4.5$

Рассмотрим треугольник АСН, прямоугольный,

Отсюда,

$AC= \sqrt{6.75 + 20.25 } = \sqrt{27} =3\sqrt{3}$

б) Периметр треугольника равен сумме сторон,

$P = 5 + 6 + 3 = 14$

в)Площадь треугольника равна половине произведения АВ на НС и на SinB

$S= \frac{1}{2} \times AB \times HC \sin( \beta )$

$S= \frac{6 \times 3}{2} \times \sin(60) = 9 \frac{ \sqrt{3} }{2} = 4.5 \sqrt{3}$

или

$S= \frac{1}{2} \times AH \times BC = \\ = 3 \times 2.6 = 7.8$

г) Радиус окружности можно вывести из формулы

$S= \frac{abc}{4 R }$

$4R = \frac{abc}{S } \\ 4R = \frac{90}{7.8} = 11.53$

$R = \frac{11.53}{4} = 2.88$

Answer 2

Task 1: Finding the length of the third side of the triangle

To find the length of the third side of the triangle, we can use the Law of Cosines. The Law of Cosines states that in a triangle with sides a, b, and c, and angle C opposite side c, the following equation holds:

c^2 = a^2 + b^2 - 2ab * cos(C)

In this case, we have two sides of the triangle, a = 3 cm and b = 6 cm, and the angle between them, C = 60°. Let's calculate the length of the third side, c:

c^2 = 3^2 + 6^2 - 2 * 3 * 6 * cos(60°)

c^2 = 9 + 36 - 36 * 0.5

c^2 = 9 + 36 - 18

c^2 = 27

Taking the square root of both sides, we get:

c = √27

c ≈ 5.196 cm

Therefore, the length of the third side of the triangle is approximately 5.196 cm.

Task 2: Finding the perimeter of the triangle

The perimeter of a triangle is the sum of the lengths of its three sides. In this case, we know the lengths of two sides: 3 cm and 6 cm. We also found the length of the third side to be approximately 5.196 cm. Let's calculate the perimeter:

Perimeter = 3 cm + 6 cm + 5.196 cm

Perimeter ≈ 14.196 cm

Therefore, the perimeter of the triangle is approximately 14.196 cm.

Task 3: Finding the area of the triangle

To find the area of the triangle, we can use the formula for the area of a triangle given two sides and the included angle:

Area = (1/2) * a * b * sin(C)

In this case, we have two sides, a = 3 cm and b = 6 cm, and the included angle, C = 60°. Let's calculate the area:

Area = (1/2) * 3 cm * 6 cm * sin(60°)

Area = (1/2) * 3 cm * 6 cm * √3/2

Area = 9 cm^2 * √3/2

Area ≈ 15.588 cm^2

Therefore, the area of the triangle is approximately 15.588 cm^2.

Task 4: Finding the radius of the circumcircle of the triangle

The circumcircle of a triangle is a circle that passes through all three vertices of the triangle. To find the radius of the circumcircle, we can use the formula:

Radius = (a * b * c) / (4 * Area)

In this case, we know the lengths of the three sides: a = 3 cm, b = 6 cm, and c ≈ 5.196 cm. We also found the area of the triangle to be approximately 15.588 cm^2. Let's calculate the radius:

Radius = (3 cm * 6 cm * 5.196 cm) / (4 * 15.588 cm^2)

Radius = 93.528 cm^3 / 62.352 cm^2

Radius ≈ 1.5 cm

Therefore, the radius of the circumcircle of the triangle is approximately 1.5 cm.

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