Triangle ABC is regular, point O is the center of triangle ABC. OM is perpendicular to the ABC plane.

| August 1, 2021 | education

Triangle ABC is regular, point O is the center of triangle ABC. OM is perpendicular to the ABC plane. Prove that MA = MB = MC. Find: MA, if AB = 6 cm, MO = 2 cm.

Since triangle ABC is regular, its center, point O, is the point of intersection of medians, heights and bisectors. The lengths of the medians of a regular triangle are equal, and at the point of their intersection they are divided by a ratio of 2/1 starting from the vertex.

Then AO = BO = CO, and then the right-angled triangles MOA, MOB, MOC are equal in two legs, which means MA = MB = MC, which was required to be proved.

The height of an equilateral triangle is: AH = BC * √3 / 2 = 6 * √3 / 2 = 3 * √3 cm.

Then AO = 2 * √3 cm.

In a right-angled triangle MOA, according to the Pythagorean theorem, AM ^ 2 = AO ^ 2 + MO ^ 2 = 12 + 4 = 16. MA = 4 cm.

Answer: The length of the MA rib is 4 cm.



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