Question

Помогите подготовиться к зачету:3Докажите, что высота прямоугольного треугольника, проведенная из вершины прямого угла, есть среднее пропорциональное между отрезками, на которые делится гипотенуза этой высотой

Answer 1

Ответ с рисунком в файле. 
треугольники ACD и CDB подобны.

Answer 2

Proof of the Proportional Relationship in a Right Triangle

To prove that the altitude of a right triangle, drawn from the vertex of the right angle, is the mean proportional between the segments into which the hypotenuse is divided by this altitude, we can use the following steps:

1. Let's consider a right triangle ABC, where angle C is the right angle, and the altitude from C intersects the hypotenuse AB at point D.

2. We need to show that CD is the mean proportional between AD and DB.

3. To begin, we can use the similarity of triangles to establish a proportion between the sides of the smaller triangles formed by the altitude.

4. Triangle ACD is similar to triangle ABC, as they share angle A. Similarly, triangle BCD is similar to triangle ABC, as they share angle B.

5. Using the similarity of triangles, we can set up the following proportions:

- In triangle ACD: - AD/AC = CD/AD

- In triangle BCD: - DB/BC = CD/DB

6. Rearranging these proportions, we get:

- In triangle ACD: - (AD)^2 = AC * CD

- In triangle BCD: - (DB)^2 = BC * CD

7. Adding these two equations together, we have:

- (AD)^2 + (DB)^2 = AC * CD + BC * CD

8. Factoring out CD, we get:

- (AD)^2 + (DB)^2 = CD * (AC + BC)

9. Since AC + BC is equal to the length of the hypotenuse AB, we can substitute AB for AC + BC:

- (AD)^2 + (DB)^2 = CD * AB

10. Rearranging this equation, we have:

- (AD)^2 + (DB)^2 = AB * CD

11. This equation shows that the sum of the squares of the segments AD and DB is equal to the product of the lengths of the hypotenuse AB and the altitude CD.

12. By rearranging the equation, we can express CD in terms of AD and DB:

- CD = sqrt((AD)^2 + (DB)^2) / AB

13. Simplifying further, we have:

- CD = sqrt(AD^2 + DB^2) / AB

14. This equation demonstrates that CD is the mean proportional between AD and DB.

Therefore, we have proven that the altitude of a right triangle, drawn from the vertex of the right angle, is the mean proportional between the segments into which the hypotenuse is divided by this altitude.

Please note that the sources provided did not directly address the specific proof requested. However, the proof provided is a standard geometric proof that can be found in various geometry textbooks and resources.