In a right-angled triangle ABC, the angle B is 30 °. The vertex of the right angle C is connected by a segment

| January 1, 2021 | education

In a right-angled triangle ABC, the angle B is 30 °. The vertex of the right angle C is connected by a segment to the point M, which belongs to the hypotenuse. The AMC angle is 60 °. Prove that CM is the median of a triangle.

1.∠ВМС = 180 ° – ∠АМС = 180 ° – 60 ° = 120 °.
2.∠ВСM = 180 ° – ∠МВС – ∠ ВМС = 180 ° – 30 ° – 120 ° = 30 °.
3. The angles at the side of the BC are equal, that is, ΔСВМ is isosceles. ВM = CM.
4. ∠АСМ = 90 ° – ∠ВСМ = 90 ° – 30 ° = 60 °.
5.∠CAM = 180 ° – ∠АСМ – ∠ВМС = 180 ° – 60 ° – 60 ° = 60 °.
6. In ΔАСМ all angles are equal. Therefore, its sides are also equal: AM, CM and AC.
7. AM = CM, and CM = BM. Hence, AM = BM, that is, the CM segment divides the AB side in half. And this means that SM is the median, which is what was required to be proved.



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