Question

Докажите использая метод от противного, что если стороны AB и BC треугольника ABC не равны, то его медиана BD не является его высотой

Answer 1

Медиана является высотой, только когда треугольник равнобедренный. Но так как треугольник не равнобедренный, медиана- не высота.

Answer 2

Proof that if sides AB and BC of triangle ABC are not equal, then its median BD is not its altitude

To prove this statement using the method of contradiction, we assume that the sides AB and BC of triangle ABC are not equal, but its median BD is its altitude.

Let's denote the midpoint of side AC as M. According to the definition of a median, BD passes through M.

Now, let's consider triangle ABC. Since BD is the median, it divides side AC into two equal segments, AM and MC.

If BD is also the altitude, it should be perpendicular to side AC. This means that angle BDM should be a right angle.

However, since AB and BC are not equal, triangle ABC is not an isosceles triangle. Therefore, angle B is not equal to angle C.

Now, let's consider triangle BDM. We have angle BDM as a right angle, and we know that angle B is not equal to angle C.

According to the Pythagorean theorem, in a right-angled triangle, the square of the length of the hypotenuse (BD) is equal to the sum of the squares of the lengths of the other two sides (BM and DM).

If BD is the altitude, then BM and DM should be equal. However, since angle B is not equal to angle C, the lengths of BM and DM cannot be equal.

This contradicts our assumption that BD is both the median and the altitude of triangle ABC.

Therefore, we can conclude that if sides AB and BC of triangle ABC are not equal, then its median BD is not its altitude.