Question

Легкие 40 балов, помогите решить.В прямоугольном треугольнике один из катетов равен 4, а медиана, которая к нему проведена равна 3. Найти радиус круга круга, который описан вокруг треугольника.Желательно с объяснением, спасибо.

Answer 1

ΔABC,<C=90,BC=4см,АМ-медиана,АМ=3см
Медиана делит сторону ВС пополам,следовательно МС=МВ=2см
Тогда по теореме Пифагора AC=√(AM²-MC²)=√(9-4)=√5см и
AB=√(AC²+BC²)=√(16+5)=√21
Гипотенуза АВ -диаметр описанной окружности около треугольника.
Значит радиус равен √21/2см

Answer 2

Problem Analysis

We are given a right triangle with one of the legs equal to 4 and the median drawn to that leg equal to 3. We need to find the radius of the circle circumscribed around the triangle.

Solution

To solve this problem, we can use the relationship between the sides of a right triangle and the radius of the circumscribed circle.

Let's denote the hypotenuse of the right triangle as c, one of the legs as a, and the other leg as b. The median drawn to the leg of length a is equal to half the length of the hypotenuse c.

We are given that a = 4 and the median drawn to a is equal to 3. Therefore, we have:

a = 4 c/2 = 3

We can use the Pythagorean theorem to relate the sides of the right triangle:

a^2 + b^2 = c^2

Substituting the given values, we have:

4^2 + b^2 = c^2

Simplifying, we get:

16 + b^2 = c^2

Since the median drawn to a is equal to half the length of the hypotenuse c, we can write:

c = 2 * (c/2) = 2 * 3 = 6

Substituting this value into the equation, we have:

16 + b^2 = 6^2 16 + b^2 = 36 b^2 = 36 - 16 b^2 = 20 b = sqrt(20)

Now, we have the lengths of all three sides of the right triangle: a = 4, b = sqrt(20), and c = 6.

The radius of the circumscribed circle can be found using the formula:

radius = (a * b * c) / (4 * area)

where area is the area of the right triangle.

The area of a right triangle can be found using the formula:

area = (1/2) * a * b

Substituting the given values, we have:

area = (1/2) * 4 * sqrt(20)

Simplifying, we get:

area = 2 * sqrt(20)

Now, we can substitute the values of a, b, c, and area into the formula for the radius:

radius = (4 * sqrt(20) * 6) / (4 * 2 * sqrt(20))

Simplifying, we get:

radius = 6 / 2 radius = 3

Therefore, the radius of the circle circumscribed around the triangle is 3.

Answer

The radius of the circle circumscribed around the triangle is 3.