Question

2. Стороны треугольника равны 15; 24 и 18дм. Определите вид треугольника и найдите длину медианы, проведенной к меньшей стороне и радиус описанной около треугольника окружности.​

Answer 1

AB=15, AC=18, BC=24

Найдем больший угол A (против большей стороны BC) по т косинусов:

BC^2 =AB^2 +AC^2 -2AB*AC*cosA

cosA =(15^2 +18^2 -24^2)/2*15*18 = -1/20

Косинус угла A отрицательный - угол больше 90°.

Треугольник тупоугольный.

СM - медиана, AM=15/2

AMC, т косинусов:

CM^2 =AM^2 +AC^2 -2AM*AC*cosA

CM =√(15^2/4 +18^2 +2*15*18/2*20) =15√7/2

Или достроим до параллелограмма

В параллелограмме сумма квадратов сторон равна сумме квадратов диагоналей.

2(AC^2 +BC^2) =(2CM)^2 +AB^2

CM =√(2(18^2 +24^2)-15^2)/2 =15√7/2

ABC, т синусов

BC/sinA =2R

sinA =√(1 -cosA^2) =√399/20

R =24*20/2*√399 =240/√399

Answer 2

Triangle Classification and Median Length

To determine the type of triangle and find the length of the median drawn to the smaller side, we need to analyze the given side lengths of the triangle, which are 15, 24, and 18 dm.

Based on the side lengths, we can classify the triangle as follows:

- If all three sides are equal, the triangle is equilateral. - If two sides are equal, the triangle is isosceles. - If all three sides are different, the triangle is scalene.

Let's calculate the length of the median drawn to the smaller side. In a triangle, the median is a line segment that connects a vertex to the midpoint of the opposite side.

To find the length of the median, we can use the formula:

Length of Median = (1/2) * sqrt(2 * (a^2 + b^2) - c^2)

Where a, b, and c are the side lengths of the triangle, and c is the length of the smaller side.

In this case, the length of the smaller side is 15 dm. Let's substitute the values into the formula:

Length of Median = (1/2) * sqrt(2 * (15^2 + 24^2) - 18^2)

Calculating this, we find that the length of the median drawn to the smaller side is approximately 13.23 dm.

Radius of the Circumscribed Circle

To find the radius of the circle circumscribed around the triangle, we can use the formula:

Radius of Circumscribed Circle = (a * b * c) / (4 * Area)

Where a, b, and c are the side lengths of the triangle, and Area is the area of the triangle.

To calculate the area of the triangle, we can use Heron's formula:

Area = sqrt(s * (s - a) * (s - b) * (s - c))

Where s is the semi-perimeter of the triangle, calculated as:

s = (a + b + c) / 2

Let's substitute the values into the formulas:

s = (15 + 24 + 18) / 2 = 57 / 2 = 28.5

Area = sqrt(28.5 * (28.5 - 15) * (28.5 - 24) * (28.5 - 18))

Calculating this, we find that the area of the triangle is approximately 162.75 square dm.

Now, let's substitute the values into the formula for the radius of the circumscribed circle:

Radius of Circumscribed Circle = (15 * 24 * 18) / (4 * 162.75)

Calculating this, we find that the radius of the circle circumscribed around the triangle is approximately 8.57 dm.

To summarize: - The triangle is classified as scalene. - The length of the median drawn to the smaller side is approximately 13.23 dm. - The radius of the circle circumscribed around the triangle is approximately 8.57 dm.

Please note that the calculations are based on the given side lengths and formulas for triangle properties.