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Geometry and Trigonometry Difficulty: Hard

Right triangle $A B C$ is shown. What is the value of $\tan A$ ?

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Explanation

Choice C is correct. In the triangle shown, the measure of angle $B$ is $30 degree$ and angle $C$ is a right angle, which means that it has a measure of $90 degree$ . Since the sum of the angles in a triangle is equal to $180 degree$ , the measure of angle $A$ is equal to $180 ° - (30 + 90) °$ , or $60 degree$ . In a right triangle whose acute angles have measures $30 degree$ and $60 degree$ , the lengths of the legs can be represented by the expressions $x$ , $x StartRoot 3 EndRoot$ , and $2 x$ , where $x$ is the length of the leg opposite the angle with measure $30 degree$ , $x StartRoot 3 EndRoot$ is the length of the leg opposite the angle with measure $60 degree$ , and $2 x$ is the length of the hypotenuse. In the triangle shown, the hypotenuse has a length of $54$ . It follows that $2 x equals 54$ , or $x equals 27$ . Therefore, the length of the leg opposite angle $B$ is $27$ and the length of the leg opposite angle $A$ is $27 StartRoot 3 EndRoot$ . The tangent of an acute angle in a right triangle is defined as the ratio of the length of the leg opposite the angle to the length of the leg adjacent to the angle. The length of the leg opposite angle $A$ is $27 StartRoot 3 EndRoot$ and the length of the leg adjacent to angle $A$ is $27$ . Therefore, the value of $tan A$ is $StartFraction 27 StartRoot 3 EndRoot Over 27 EndFraction$ , or $StartRoot 3 EndRoot$ .

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

Choice B is incorrect. This is the value of $StartFraction 1 Over tangent upper A EndFraction$ , not the value of $tan A$ .

Choice D is incorrect. This is the length of the leg opposite angle $A$ , not the value of $tan A$ .