### Instruction: The altitude of an equilateral triangle is $\sqrt6$ units.  What is the area of the triangle, in square units? Express your answer in simplest radical form.### Response: I know that an equilateral triangle has three equal sides and three equal angles of 60 degrees each.
I also know that the altitude of an equilateral triangle is a perpendicular bisector of the base, so it divides the base into two congruent segments and forms two right triangles.
I can label the base of the triangle as 2x and the altitude as h, where h = sqrt6.
Then, using the Pythagorean theorem, I can write an equation for one of the right triangles: x^2 + h^2 = (2x)^2.
Simplifying the equation, I get x^2 + 6 = 4x^2, or 3x^2 = 6, or x^2 = 2, or x = sqrt2.
So the base of the equilateral triangle is 2x = 2sqrt2.
Now, to find the area of the triangle, I can use the formula A = (1/2)bh, where b is the base and h is the altitude.
Plugging in the values I found, I get A = (1/2)(2sqrt2)(sqrt6) = sqrt12.
I can simplify sqrt12 by factoring out a perfect square: sqrt12 = sqrt(4*3) = sqrt4 * sqrt3 = 2sqrt3.
So the area of the triangle is 2sqrt3 square units.