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Geometry and Trigonometry Difficulty: Hard

A right triangle has sides of length $2 StartRoot 2 EndRoot$ , $6 StartRoot 2 EndRoot$ , and $StartRoot 80 EndRoot$ units. What is the area of the triangle, in square units?

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Explanation

Choice B is correct. The area, $A$ , of a triangle can be found using the formula $A = \frac{1}{2} b h$ , where $b$ is the length of the base of the triangle and $h$ is the height of the triangle. It's given that the triangle is a right triangle. Therefore, its base and height can be represented by the two legs. It’s also given that the triangle has sides of length $2 StartRoot 2 EndRoot$ , $6 StartRoot 2 EndRoot$ , and $StartRoot 80 EndRoot$ units. Since $StartRoot 80 EndRoot$ units is the greatest of these lengths, it's the length of the hypotenuse. Therefore, the two legs have lengths $2 StartRoot 2 EndRoot$ and $6 StartRoot 2 EndRoot$ units. Substituting these values for $b$ and $h$ in the formula $A = \frac{1}{2} b h$ gives $upper A equals one half left parenthesis 2 StartRoot 2 EndRoot right parenthesis left parenthesis 6 StartRoot 2 EndRoot right parenthesis$ , which is equivalent to $upper A equals 6 StartRoot 4 EndRoot$ square units, or $upper A equals 12$ square units.

Choice A is incorrect. This expression represents the perimeter, rather than the area, of the triangle.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.