Question

- У трикутнику одна з медіан перпендикулярна одній з бісектрис. Доведіть, що одна з сторін цього трикутника вдвічі більше другої.​

Answer 1

Ответ:

Объяснение: У трикутнику одна з медіан перпендикулярна одній з бісектрис.

Answer 2

The Relationship Between Medians and Bisectors in a Triangle

In a triangle, a median is a line segment that connects a vertex to the midpoint of the opposite side. A bisector is a line segment that divides an angle into two equal parts. The question states that one of the medians is perpendicular to one of the bisectors in the triangle. We need to prove that one side of the triangle is twice as long as the other side.

To prove this, let's consider the given information and use some properties of medians and bisectors in a triangle.

Proof:

Let's assume we have a triangle ABC, where AD is a median and BE is a bisector. We need to prove that one side of the triangle is twice as long as the other side.

1. Given: AD is perpendicular to BE.

2. Let's consider the triangle ABC and draw the median AD from vertex A to the midpoint of side BC. Let's also draw the bisector BE from vertex B, dividing angle B into two equal parts.

3. Since AD is a median, it divides side BC into two equal parts. Let's denote the length of BD as x and the length of DC as x as well.

4. Since BE is a bisector, it divides angle B into two equal parts. Let's denote the measure of angle ABD as α and the measure of angle DBC as α as well.

5. Since AD is perpendicular to BE, angle ADB is a right angle.

6. In triangle ADB, we have a right angle at D and two equal angles (α and α) at A and B. Therefore, triangle ADB is an isosceles triangle.

7. In an isosceles triangle, the sides opposite to the equal angles are equal. Therefore, AB = AD.

8. In triangle BDC, we have two equal sides (BD = DC) and two equal angles (α and α) at B and C. Therefore, triangle BDC is an isosceles triangle.

9. In an isosceles triangle, the angles opposite to the equal sides are equal. Therefore, angle BDC = angle BCD.

10. Since angle BDC = angle BCD, triangle BDC is an isosceles triangle.

11. In an isosceles triangle, the altitude from the vertex angle bisects the base. Therefore, DE is the altitude from vertex D in triangle BDC, and it bisects BC.

12. Since DE bisects BC, we can conclude that BD = DC.

13. Combining the results from steps 7 and 12, we have AB = AD = BD = DC.

14. From step 13, we can conclude that one side of the triangle (AB) is twice as long as the other side (BC).

Therefore, we have proved that one side of the triangle is twice as long as the other side, given that one of the medians is perpendicular to one of the bisectors.

Note: The proof assumes that the given triangle is non-degenerate and that the perpendicularity and bisecting properties hold true.