Question

Із центра О вписаного в прямокутний трикутник АБС кола до площини цього трикутника про- ведено перпендикуляр Ом завдовжки корінь 8 см. Знайдіть довжину перпендикуляра, проведеногоз точки M до меншого катета трикутника АВС, якщо його катети дорівнюють 3 см і 4 см. СРОЧНО!!! ​

Answer 1

Відповідь: 3 см

Пояснення: фото

Answer 2

To find the length of the perpendicular drawn from point M to the smaller cathetus of triangle ABC, we can use the properties of similar triangles. Let's break down the problem step by step.

Given Information:

- Triangle ABC is a right-angled triangle with the right angle at point C. - The cathetus of triangle ABC have lengths of 3 cm and 4 cm. - A circle is inscribed in triangle ABC, with the center O. - A perpendicular line OM is drawn from the center O of the inscribed circle to the plane of triangle ABC. - The length of OM is √8 cm.

Solution:

1. Let's denote the length of the perpendicular drawn from point M to the smaller cathetus of triangle ABC as x cm. 2. Since OM is perpendicular to the plane of triangle ABC, it is also perpendicular to the smaller cathetus. 3. We can form two similar triangles: triangle OMC and triangle ABC. 4. The ratio of corresponding sides of similar triangles is equal. Therefore, we can set up the following proportion:

MC / AC = OM / AB

Substituting the given values:

x / 3 = √8 / 5

5. Cross-multiplying the proportion:

5x = 3√8

6. Solving for x:

x = (3√8) / 5

Simplifying the expression:

x = (3 * 2√2) / 5

x = (6√2) / 5

Therefore, the length of the perpendicular drawn from point M to the smaller cathetus of triangle ABC is (6√2) / 5 cm.

Please note that the solution provided is based on the given information and the assumption that the triangle is a right-angled triangle with the given cathetus lengths.