Question

4. (5Б) В равнобедренном треугольнике ABC с основанием АС проведена медиана ВД. Точка Р- середина стороны BA, точка K — середина стороны ВС. Докажите равенство треугольников ВДР и ВДК. -​

Answer 1

Ответ:

ΔАВС ,  АВ=ВС  ,  АР=РВ ,  ВК=КС  .

 Так как АВ=ВС , то и половины этих сторон равны между собой: АР=РВ=ВК=КС .

Рассм. ΔВДР и ΔВДС . У них ВР=ВК , ВД - общая сторона и ∠РВД=∠КВД , так как медиана равнобедр. тр-ка , проведённая к основанию, является ещё и биссектрисой .

Значит, ΔВДР=ΔВДС по 1 признаку равенства треугольников .

Answer 2

Proof of Triangle Congruence

To prove that triangles VDR and VDK are congruent, we can use the congruence criterion for triangles. In this case, we will use the Side-Angle-Side (SAS) criterion.

Given: - Triangle ABC is an isosceles triangle with base AC. - Median BD is drawn from vertex B, intersecting AC at point R. - Point P is the midpoint of side BA. - Point K is the midpoint of side BC.

To prove: - Triangle VDR ≅ Triangle VDK

Proof:

1. Since triangle ABC is an isosceles triangle with base AC, we know that the base angles at A and C are congruent. This is because in an isosceles triangle, the angles opposite the congruent sides are congruent.

2. Since P is the midpoint of side BA, we can conclude that BP = PA. Similarly, since K is the midpoint of side BC, we have BK = KC.

3. Since R is the intersection point of median BD and base AC, we can conclude that AR = RC. This is because the medians of a triangle divide each other into segments of equal length.

4. Now, let's consider triangles VDR and VDK. We can see that: - VR is common to both triangles. - DR = DK (since R and K are midpoints of sides BA and BC, respectively). - RD = RK (since AR = RC).

5. Therefore, by the Side-Angle-Side (SAS) criterion, we can conclude that triangles VDR and VDK are congruent.

Thus, we have proved that triangle VDR ≅ triangle VDK using the Side-Angle-Side (SAS) criterion.

Please note that the sources provided did not contain specific information related to this geometric problem. The proof provided is based on the properties and definitions of isosceles triangles, medians, and midpoints.