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Geometry and Trigonometry Difficulty: Hard

Triangle $A B C$ is similar to triangle $D E F$ , where $A$ corresponds to $D$ and $C$ corresponds to $F$ . Angles $C$ and $F$ are right angles. If $tangent left parenthesis upper A right parenthesis equals StartRoot 3 EndRoot$ and $upper D upper F equals 125$ , what is the length of $\bar{D E}$ ?

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Explanation

Choice D is correct. Corresponding angles in similar triangles have equal measures. It's given that triangle $A B C$ is similar to triangle $D E F$ , where $A$ corresponds to $D$ , so the measure of angle $A$ is equal to the measure of angle $D$ . Therefore, if $tangent left parenthesis upper A right parenthesis equals StartRoot 3 EndRoot$ , then $tangent left parenthesis upper D right parenthesis equals StartRoot 3 EndRoot$ . It's given that angles $C$ and $F$ are right angles, so triangles $A B C$ and $D E F$ are right triangles. The adjacent side of an acute angle in a right triangle is the side closest to the angle that is not the hypotenuse. It follows that the adjacent side of angle $D$ is side $D F$ . The opposite side of an acute angle in a right triangle is the side across from the acute angle. It follows that the opposite side of angle $D$ is side $E F$ . The tangent of an acute angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side. Therefore, $\tan (D) = \frac{E F}{D F}$ . If $upper D upper F equals 125$ , the length of side $E F$ can be found by substituting $StartRoot 3 EndRoot$ for $\tan (D)$ and $125$ for $D F$ in the equation $\tan (D) = \frac{E F}{D F}$ , which yields $\sqrt{3} = \frac{E F}{125}$ . Multiplying both sides of this equation by $125$ yields $125 StartRoot 3 EndRoot equals upper E upper F$ . Since the length of side $E F$ is $StartRoot 3 EndRoot$ times the length of side $D F$ , it follows that triangle $D E F$ is a special right triangle with angle measures $30 degree$ , $60 degree$ , and $90 degree$ . Therefore, the length of the hypotenuse, $\bar{D E}$ , is $2$ times the length of side $D F$ , or $upper D upper E equals 2 left parenthesis upper D upper F right parenthesis$ . Substituting $125$ for $D F$ in this equation yields $upper D upper E equals 2 left parenthesis 125 right parenthesis$ , or $upper D upper E equals 250$ . Thus, if $tangent left parenthesis upper A right parenthesis equals StartRoot 3 EndRoot$ and $upper D upper F equals 125$ , the length of $\bar{D E}$ is $250$ .

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

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

Choice C is incorrect. This is the length of $\bar{E F}$ , not $\bar{D E}$ .