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Triangle $A B C$ is similar to triangle $D E F$ , where angle $A$ corresponds to angle $D$ and angles $C$ and $F$ are right angles. The length of $\bar{A B}$ is $2.9$ times the length of $\bar{D E}$ . If $\tan A = \frac{21}{20}$ , what is the value of $\sin D$ ?

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Explanation

The correct answer is $StartFraction 21 Over 29 EndFraction$ . It's given that triangle $A B C$ is similar to triangle $D E F$ , where angle $A$ corresponds to angle $D$ and angles $C$ and $F$ are right angles. In similar triangles, the tangents of corresponding angles are equal. Therefore, if $\tan A = \frac{21}{20}$ , then $\tan D = \frac{21}{20}$ . In a right triangle, the tangent of an acute angle is the ratio of the length of the leg opposite the angle to the length of the leg adjacent to the angle. Therefore, in triangle $D E F$ , if $\tan D = \frac{21}{20}$ , the ratio of the length of $\bar{E F}$ to the length of $\bar{D F}$ is $StartFraction 21 Over 20 EndFraction$ . If the lengths of $\bar{E F}$ and $\bar{D F}$ are $21$ and $20$ , respectively, then the ratio of the length of $\bar{E F}$ to the length of $\bar{D F}$ is $StartFraction 21 Over 20 EndFraction$ . In a right triangle, the sine of an acute angle is the ratio of the length of the leg opposite the angle to the length of the hypotenuse. Therefore, the value of $\sin D$ is the ratio of the length of $\bar{E F}$ to the length of $\bar{D E}$ . The length of $\bar{D E}$ can be calculated using the Pythagorean theorem, which states that if the lengths of the legs of a right triangle are $a$ and $b$ and the length of the hypotenuse is $c$ , then $a^{2} + b^{2} = c^{2}$ . Therefore, if the lengths of $\bar{E F}$ and $\bar{D F}$ are $21$ and $20$ , respectively, then ${(21)}^{2} + {(20)}^{2} = {(D E)}^{2}$ , or $841 equals left parenthesis upper D upper E right parenthesis squared$ . Taking the positive square root of both sides of this equation yields $29 equals upper D upper E$ . Therefore, if the lengths of $\bar{E F}$ and $\bar{D F}$ are $21$ and $20$ , respectively, then the length of $\bar{D E}$ is $29$ and the ratio of the length of $\bar{E F}$ to the length of $\bar{D E}$ is $StartFraction 21 Over 29 EndFraction$ . Thus, if $\tan A = \frac{21}{20}$ , the value of $\sin D$ is $StartFraction 21 Over 29 EndFraction$ . Note that 21/29, .7241, and 0.724 are examples of ways to enter a correct answer.