Question

На медиане МС равнобедренном треугольнике МРК с основанием РК отмечена точка В. Докажите, что а) ∆МВК = ∆МВР; б) ∆КВС = ∆МВС.

Answer 1

А) В равнобедренном треугольнике медиана, проведённая к основанию, является и биссектрисой, и высотой. Тогда ∠PMB = ∠KMB.
PM = MK - как боковые стороны.
BM - общая сторона.
Значит, ∆MBK = ∆MBP - по I признаку.

б) Т.к. MC - медиана, то PC = CК, а т.к. МС - и высота, то ∠PCM = ∠KCM = 90°.
BC - общая сторона.
Значит, ∆KBC = ∆PBC - по I признаку.

Answer 2

По условию треугольник ПМК равнобедренный с основанием ПК. Медиана в таком треугольнике равно биссектрисе и высоте. Таким образом угол ПМБ=углу КМБю Треугольники подобны и равны по признаку равенства двух сторон МП=МК , Мб - общая и углу между нимим.

Answer 3

Mediana of an Isosceles Triangle

To prove that ∆МВК = ∆МВР and ∆КВС = ∆МВС, we need to use the properties of an isosceles triangle and the definition of a median.

Let's start by defining the given triangle and its medians:

- Triangle МРК is an isosceles triangle with base РК. - Point В is the midpoint of side РК. - МВ is one of the medians of triangle МРК.

Now, let's prove the two statements:

Proof: ∆МВК = ∆МВР

To prove that ∆МВК = ∆МВР, we need to show that they have the same three sides.

1. Side МВ: Since В is the midpoint of side РК, we can conclude that МВ is the median of triangle МРК. The median of a triangle connects a vertex to the midpoint of the opposite side. Therefore, МВ is common to both triangles ∆МВК and ∆МВР.

2. Side ВК: In triangle ∆МВК, side ВК is equal to side ВК itself.

3. Side ВР: In triangle ∆МВР, side ВР is equal to side ВР itself.

Since all three sides of ∆МВК are equal to the corresponding sides of ∆МВР, we can conclude that ∆МВК = ∆МВР.

Proof: ∆КВС = ∆МВС

To prove that ∆КВС = ∆МВС, we also need to show that they have the same three sides.

1. Side КВ: In triangle ∆КВС, side КВ is equal to side КВ itself.

2. Side ВС: In triangle ∆КВС, side ВС is equal to side ВС itself.

3. Side МВ: In triangle ∆МВС, side МВ is equal to side МВ itself.

Since all three sides of ∆КВС are equal to the corresponding sides of ∆МВС, we can conclude that ∆КВС = ∆МВС.

Therefore, we have proved that ∆МВК = ∆МВР and ∆КВС = ∆МВС.

Please let me know if you have any further questions!