Question

1. У трикутнику АВС провели медіану АМ. Із точок В і С на пряму АМ опустили перпендикуляри ВК і CN. Доведіть, що KM=NM.

Answer 1

Ответ:

бо за теоремою про перпендикуляр КМ=NM

Answer 2

Proof of KM=NM in Triangle ABC

To prove that KM=NM in triangle ABC, where AM is the median and perpendiculars BK and CN are dropped from points B and C onto line AM, we can use the properties of medians and perpendiculars in a triangle.

First, let's consider the properties of medians in a triangle. The median of a triangle divides the opposite side into two segments of equal length. This means that in triangle ABC, the median AM divides side BC into two equal segments, BM and MC.

Next, we'll consider the properties of perpendiculars in a triangle. When a perpendicular is dropped from a vertex of a triangle to the opposite side, it divides the opposite side into two segments. These segments are proportional to the adjacent sides of the triangle.

Now, let's apply these properties to prove that KM=NM.

Since AM is the median of triangle ABC, it divides side BC into two equal segments, BM and MC. Therefore, BM=MC.

When perpendiculars BK and CN are dropped from points B and C onto line AM, they divide AM into three segments: AK, KM, and MN.

Given that BM=MC, we can conclude that the perpendiculars BK and CN also divide AM into two equal segments, making KM=NM.

Therefore, we have proved that KM=NM in triangle ABC.

This completes the proof.