Question

У прямокутний трикутник, периметр якого 24 см, вписано коло, точка дотику ділить гіпотенузу на відрізки 4см і 5см. Знайдіть радіус цього кола.

Answer 1

Ответ:

3см

Объяснение:

.........................

Answer 2

Problem Analysis

We are given a right triangle with a perimeter of 24 cm and an inscribed circle. The point of tangency divides the hypotenuse into segments of 4 cm and 5 cm. We need to find the radius of this circle.

Solution

To solve this problem, we can use the properties of a right triangle and the inscribed circle.

Let's denote the legs of the right triangle as a and b, and the hypotenuse as c. We are given that the perimeter of the triangle is 24 cm, so we can write the equation:

a + b + c = 24 [[1]]

We are also given that the point of tangency divides the hypotenuse into segments of 4 cm and 5 cm. Let's denote the distance from the point of tangency to the vertex of the right angle as h. Using the properties of similar triangles, we can write the following equations:

h/a = r/c [[2]] h/b = r/c [[3]]

where r is the radius of the inscribed circle.

From equations [[2]] and [[3]], we can express h in terms of a, b, and r:

h = (ar)/c [[4]] h = (br)/c [[5]]

Setting equations [[4]] and [[5]] equal to each other, we get:

(ar)/c = (br)/c

Simplifying, we find:

a = b [[6]]

From equation [[6]], we can conclude that the right triangle is an isosceles right triangle.

Now, let's substitute equation [[6]] into equation [[1]]:

a + a + c = 24

Simplifying, we find:

2a + c = 24 [[7]]

We can solve equations [[6]] and [[7]] simultaneously to find the values of a and c.

Solving equation [[6]] for a, we get:

a = b = (24 - c)/2

Substituting this into equation [[7]], we have:

2((24 - c)/2) + c = 24

Simplifying, we find:

24 - c + c = 24

24 = 24

This equation is always true, which means that any value of c will satisfy it. Therefore, there are infinitely many solutions for the lengths of the legs of the right triangle.

However, we can still find the radius of the inscribed circle using the given information. Let's substitute a = 4 cm and b = 5 cm into equation [[2]]:

(h/4) = r/c

Simplifying, we find:

h = (4r)/c [[8]]

We are also given that h = 5 cm. Substituting this into equation [[8]], we have:

5 = (4r)/c

Simplifying, we find:

r = (5c)/4 [[9]]

From equation [[9]], we can see that the radius of the inscribed circle depends on the value of c. Therefore, we cannot determine the exact value of the radius without knowing the length of the hypotenuse.

In conclusion, the radius of the inscribed circle in the right triangle depends on the length of the hypotenuse, which is not given in the problem statement.