Given: ABC is an equilateral triangle. Line a is perpendicular to plane ABC. AB = 2√3, MD = 4. Find MC

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And based on the condition, it is known that D is the middle of the segment AB. And CD is the median of an equilateral triangle, hence the height, according to the property of an equilateral triangle.

Means:

AD = AB / 2 = 2√3 / 2 = √3;

AC = AB = 2√3.

Consider ΔADC, where ∠ ADC = 90 °.

Let’s apply the Pythagorean theorem:

CD ^ 2 = AC ^ 2 – AD ^ 2 = (2√3) ^ 2 – (√3) ^ 2 = 12 – 3 = 9;

CD = 3.

The condition says that the straight line a is perpendicular to the plane ABC, which means that it is perpendicular to any straight line lying in this plane, i.e. ∠ MDC = 90 °.

We apply the Pythagorean theorem to Δ MDC:

MC = √ (MD ^ 2 + CD ^ 2) = √ (16 + 9) = √25 = 5.