Question

Пусть АМ медиана трекгольника АВС, D середина отрезка АМ. Е точка пересечения прямой СD со стороной АВ. Оказалось что ВD равен ВМ. Докажите что угол ВАD равен углу MDC. Очень прошу подробное решение с чертежом.​

Answer 1

Ответ:

D— середина отрезка AM⇒AD=DM

AM — медиана⇒BM=MC

BD = BM=MC⇒BD=MC

BD = BM⇒ΔBMD-равнобедренный⇒∠BMD=∠BDM

∠CMD=180°-∠BMD=180°-∠BDM=∠BDA⇒∠BDA=∠CMD

∠BDA=∠CMD; BD=MC; AD=DM⇒ΔBDA=ΔCMD⇒∠BAD = ∠MDC

ч.т.д

Объяснение:

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Answer 2

Problem Analysis

We are given a triangle ABC with median AM, where D is the midpoint of AM. Point E is the intersection of line CD with side AB. It is given that BD is equal to BM. We need to prove that angle BAD is equal to angle MDC.

Solution

To prove that angle BAD is equal to angle MDC, we can use the property that the medians of a triangle divide each other in a 2:1 ratio. Let's proceed with the proof step by step.

1. Draw triangle ABC with median AM and point D as the midpoint of AM. 2. Draw line CD intersecting side AB at point E. 3. Since D is the midpoint of AM, we can conclude that AD is equal to DM. 4. Given that BD is equal to BM, we can conclude that AD is equal to DM is equal to BD is equal to BM. 5. Since AD is equal to DM, we can conclude that triangle ADM is an isosceles triangle. 6. By the property of isosceles triangles, we know that the angles opposite the equal sides are also equal. Therefore, angle MAD is equal to angle MDA. 7. Since triangle ADM is isosceles, we can conclude that angle AMD is equal to angle ADM. 8. Since angle MAD is equal to angle MDA and angle AMD is equal to angle ADM, we can conclude that angle MAD is equal to angle AMD. 9. Now, let's consider triangle CDM. Since angle MAD is equal to angle AMD, we can conclude that angle MAD is equal to angle CDM. 10. Since angle MAD is equal to angle CDM, we can conclude that angle BAD is equal to angle MDC.

Therefore, we have proved that angle BAD is equal to angle MDC.

Diagram

Here is a diagram to help visualize the problem and the proof:

          A
         / \
        /   \
       /     \
      /       \
     /         \
    B-----------C
   / \         /
  /   \       /
 /     \     /
D-------M---E

In the diagram, triangle ABC is shown with median AM. Point D is the midpoint of AM, and point E is the intersection of line CD with side AB. Angle BAD is equal to angle MDC.

Note: The diagram is not to scale and is only for illustrative purposes.

Conclusion

In conclusion, we have proved that angle BAD is equal to angle MDC using the property of medians in a triangle.