Question

У трикутнику ABC проведено бісектрису BD. Відомо, що AB = 15 см, BC = 10 см. Доведіть, що BD < 12 см.

Answer 1

Вітаю.

Пояснення: фото.

Спокійної цієї ночі.

Answer 2

Given Information:

We are given a triangle ABC with a bisector BD. It is known that AB = 15 cm and BC = 10 cm. We need to prove that BD < 12 cm.

Proof:

To prove that BD < 12 cm, we can use the triangle inequality theorem. According to this theorem, in any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.

In triangle ABC, let's consider the lengths of the sides AB, BC, and AC. We know that AB = 15 cm and BC = 10 cm.

Using the triangle inequality theorem, we can write the following inequalities:

1. AB + BC > AC 2. AB + AC > BC 3. BC + AC > AB

Let's substitute the given values:

1. 15 cm + 10 cm > AC 2. 15 cm + AC > 10 cm 3. 10 cm + AC > 15 cm

Simplifying these inequalities, we get:

1. 25 cm > AC 2. AC > -5 cm (This inequality is not useful for our proof) 3. AC > 5 cm

Now, let's consider triangle ABD. We know that BD is the bisector of angle ABC. By the angle bisector theorem, we can write:

AB/BD = AC/CD

Substituting the given values, we have:

15 cm/BD = AC/CD

Simplifying, we get:

AC = (15 cm * CD) / BD

Since AC > 5 cm (from the previous inequality), we can write:

(15 cm * CD) / BD > 5 cm

Simplifying further, we have:

15 cm * CD > 5 cm * BD

Dividing both sides by 5 cm, we get:

3 cm * CD > BD

Now, let's consider triangle BCD. We know that BC = 10 cm and CD is a side of triangle BCD. Using the triangle inequality theorem again, we can write:

BC + CD > BD

Substituting the given values, we have:

10 cm + CD > BD

Simplifying, we get:

CD > BD - 10 cm

Since we have already proved that 3 cm * CD > BD, we can substitute this inequality into the previous one:

3 cm * (BD - 10 cm) > BD

Expanding, we get:

3 cm * BD - 30 cm > BD

Simplifying further, we have:

2 cm * BD > 30 cm

Dividing both sides by 2 cm, we get:

BD > 15 cm

From the previous inequality, we have also proved that BD < 12 cm. Therefore, we have a contradiction.

Hence, our assumption that BD < 12 cm is incorrect.

Therefore, we can conclude that BD is not less than 12 cm.

In conclusion, we have proved that BD is not less than 12 cm.