Question

Катеты прямоугольного треугольника равны 7 и 24. найти расстояние между центрами вписанной и описанной окружностей. пожалуйста с рисунком

Answer 1

Решение смотри в файле

Answer 2

Finding the Distance Between the Centers of the Inscribed and Circumscribed Circles of a Right Triangle

To find the distance between the centers of the inscribed and circumscribed circles of a right triangle, we need to use the properties of right triangles and circles.

Given that the lengths of the legs (or catheti) of the right triangle are 7 and 24, we can calculate the length of the hypotenuse using the Pythagorean theorem.

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

Let's calculate the length of the hypotenuse (c) using the given lengths of the legs (a = 7 and b = 24):

Step 1: Calculate the square of the length of the hypotenuse (c^2) using the Pythagorean theorem:

c^2 = a^2 + b^2

c^2 = 7^2 + 24^2

c^2 = 49 + 576

c^2 = 625

Step 2: Take the square root of both sides to find the length of the hypotenuse (c):

c = √625

c = 25

So, the length of the hypotenuse (c) is 25.

Now that we have the lengths of the sides of the right triangle, we can find the distance between the centers of the inscribed and circumscribed circles.

The distance between the centers of the inscribed and circumscribed circles is equal to the difference between the radii of the two circles.

The radius of the inscribed circle (r1) can be calculated using the formula:

r1 = (a + b - c) / 2

Substituting the values, we get:

r1 = (7 + 24 - 25) / 2

r1 = 6

The radius of the circumscribed circle (r2) is equal to half the length of the hypotenuse (c) of the right triangle:

r2 = c / 2

r2 = 25 / 2

r2 = 12.5

The distance between the centers of the inscribed and circumscribed circles is the difference between the radii:

distance = r2 - r1

distance = 12.5 - 6

distance = 6.5

Therefore, the distance between the centers of the inscribed and circumscribed circles of the right triangle is 6.5 units.

Please note that the above calculations are based on the given lengths of the legs of the right triangle.