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Triangle $ABC$ is similar to triangle $DEF$, where $A$ corresponds to $D$ and $C$ corresponds to $F$. Angles $C$ and $F$ are right angles. If $\\tan(A)=\\sqrt{3}$ and $DF=125$, what is the length of $\\overline{DE}$?

","hasPart":{"@type":"Question","typicalAgeRange":"12-22","educationalAlignment":[{"@type":"AlignmentObject","alignmentType":"educationalSubject","targetName":"SAT"},{"@type":"AlignmentObject","alignmentType":"educationalLevel","targetName":"SAT","educationalFramework":"Common Core","targetUrl":"https://classnotes.xyz/mcqs/sat-math-geometry-and-trigonometry-multiple-choice-question-36"}],"about":{"@type":"Thing","name":"Math, Geometry and Trigonometry, Duplicate"},"educationalLevel":"intermediate","eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","assesses":["Math","Geometry and Trigonometry","Duplicate"],"name":"

Triangle $ABC$ is similar to triangle $DEF$, where $A$ corresponds to $D$ and $C$ corresponds to $F$. Angles $C$ and $F$ are right angles. If $\\tan(A)=\\sqrt{3}$ and $DF=125$, what is the length of $\\overline{DE}$?

","comment":{"@type":"Comment","text":"

Choice D is correct. Corresponding angles in similar triangles have equal measures. It's given that triangle $ABC$ is similar to triangle $DEF$, where $A$ corresponds to $D$, so the measure of angle $A$ is equal to the measure of angle $D$. Therefore, if $\\tan(A)=\\sqrt{3}$, then $\\tan(D)=\\sqrt{3}$. It's given that angles $C$ and $F$ are right angles, so triangles $ABC$ and $DEF$ 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 $DF$. 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 $EF$. 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, $\\tan(D)=\\frac{EF}{DF}$. If $DF=125$, the length of side $EF$ can be found by substituting $\\sqrt{3}$ for $\\tan(D)$ and $125$ for $DF$ in the equation $\\tan(D)=\\frac{EF}{DF}$, which yields $\\sqrt{3}=\\frac{EF}{125}$. Multiplying both sides of this equation by $125$ yields $125\\sqrt{3}=EF$. Since the length of side $EF$ is $\\sqrt{3}$ times the length of side $DF$, it follows that triangle $DEF$ is a special right triangle with angle measures $30^{{}^{\\circ}},60^{{}^{\\circ}}$, and $90^{{}^{\\circ}}$. Therefore, the length of the hypotenuse, $\\overline{DE}$, is $2$ times the length of side $DF$, or $DE=2(DF)$. Substituting $125$ for $DF$ in this equation yields $DE=2(125)$, or $DE=250$. Thus, if $\\tan(A)=\\sqrt{3}$ and $DF=125$, the length of $\\overline{DE}$ 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 $\\overline{EF}$, not $\\overline{DE}$.

"},"encodingFormat":"text/markdown","text":"

Triangle $ABC$ is similar to triangle $DEF$, where $A$ corresponds to $D$ and $C$ corresponds to $F$. Angles $C$ and $F$ are right angles. If $\\tan(A)=\\sqrt{3}$ and $DF=125$, what is the length of $\\overline{DE}$?

","suggestedAnswer":[{"@type":"Answer","position":0,"encodingFormat":"text/html","text":"

$125\\frac{\\sqrt{3}}{3}$

"},{"@type":"Answer","position":1,"encodingFormat":"text/html","text":"

$125\\frac{\\sqrt{3}}{2}$

"},{"@type":"Answer","position":2,"encodingFormat":"text/html","text":"

\$125\\sqrt{3}\$

"}],"acceptedAnswer":{"@type":"Answer","position":3,"encodingFormat":"text/html","text":"

$250$

","comment":{"@type":"Comment","text":"

"},"answerExplanation":{"@type":"Comment","text":"

"}}},"url":"https://classnotes.xyz/mcqs/sat-math-geometry-and-trigonometry-multiple-choice-question-36"}

Triangle $ABC$ is similar to triangle $DEF$, where $A$ corresponds to $D$ and $C$ corresponds to $F$. Angles $C$ and $F$ are right angles. If $\tan(A)=\sqrt{3}$ and $DF=125$, what is the length of $\overline{DE}$?