An equilateral triangle has a height of 9 centimeters. The area of the triangle is k3 square centimeters. What is the value of k?

An equilateral triangle has a height of 9 centimeters. The... | SAT Right triangles and trigonometry Question, Answer Explained

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Math · Geometry and TrigonometryHard

An equilateral triangle has a height of $9$ centimeters. The area of the triangle is $k\sqrt{3}$ square centimeters. What is the value of $k$ ?

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Answer: 27

The height splits the equilateral triangle into two $30^\circ$ - $60^\circ$ - $90^\circ$ triangles, with the height opposite the $60^\circ$ angle. So half the base is $\frac{9}{\sqrt{3}} = 3\sqrt{3}$ , and the base is $6\sqrt{3}$ . The area is $\frac{1}{2}(6\sqrt{3})(9) = 27\sqrt{3}$ , so $k = 27$ .

Treating $9$ as the side length gives an area of $\frac{81\sqrt{3}}{4}$ , but $9$ is the height, which is shorter than a side.

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