Question

Діагоналі ромба дорівнюють 30 см і 40 см. Куля дотикається до всіх сторін ромба, а відстань від центра кулі до площини ромба дорівнює 18 см. Знайти радіус кулі.

Answer 1

Відповідь:

Пояснення:

Answer 2

Diagonal lengths of a rhombus

The diagonal lengths of a rhombus are given as 30 cm and 40 cm.

Sphere touching all sides of the rhombus

A sphere is touching all sides of the rhombus, and the distance from the center of the sphere to the plane of the rhombus is given as 18 cm.

Finding the radius of the sphere

To find the radius of the sphere, we can use the following steps:

1. The diagonals of a rhombus bisect each other at right angles. Therefore, the diagonals divide the rhombus into four congruent right-angled triangles. 2. Let's consider one of these right-angled triangles. The hypotenuse of the triangle is one of the diagonals of the rhombus, which has a length of 40 cm. 3. The distance from the center of the sphere to the plane of the rhombus is the height of this right-angled triangle, which is given as 18 cm. 4. We can use the Pythagorean theorem to find the length of the other two sides of the right-angled triangle. - Let's denote the length of one of the legs as a and the length of the other leg as b. - According to the Pythagorean theorem, a^2 + b^2 = c^2, where c is the length of the hypotenuse. - In this case, a^2 + 18^2 = 40^2. 5. Solving the equation, we find that a^2 = 40^2 - 18^2. 6. Taking the square root of both sides, we get a = sqrt(40^2 - 18^2). 7. Now, we have the length of one of the legs of the right-angled triangle, which is also the radius of the sphere. 8. Therefore, the radius of the sphere is sqrt(40^2 - 18^2).

Let's calculate the radius using the given values:

a = sqrt(40^2 - 18^2) = sqrt(1600 - 324) = sqrt(1276) ≈ 35.74 cm

Therefore, the radius of the sphere is approximately 35.74 cm.