Bunuel wrote:
The side of an equilateral triangle has the same length as the diagonal of a square. What is the area of the square?
(1) The height of the equilateral triangle is equal to \(6\sqrt{3}\).
(2) The area of the equilateral triangle is equal to \(36\sqrt{3}\).
Kudos for a correct solution.
MANHATTAN GMAT OFFICIAL SOLUTION:No calculation is needed to solve this problem. Both equilateral triangles and squares are regular figures— those that can change size, but never shape.
Regular figures (squares, equilaterals, circles, spheres, cubes, 45-45-90 triangles, 30-60-90 triangles, and others) are those for which you only need one measurement to know every measurement. For instance, if you have the radius of a circle, you can get the diameter, circumference, and area. If you have a 45-45-90 or 30-60-90 triangle, you only need one side to get all three. In this problem, if you have the side of an equilateral, you could get the height, area, and perimeter. If you have the side of a square, you could get the diagonal, area, and perimeter.
If you have two regular figures, as you do in this problem, and you know how they are related numerically (“the side of an equilateral triangle has the same length as the diagonal of a square”), then you can safely conclude that any measurement for either figure will give you any measurement for either figure.
The question can be rephrased as, “What is the length of any part of either figure?”
1) This gives you the height of the triangle. SUFFICIENT.
2) This gives you the area of the triangle. SUFFICIENT.
Answer: D.
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