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

The correct answer is \(\frac{21}{29}\). It's given that triangle ABC is similar to triangle DEF, 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 DEF, if \(\tan D=\frac{21}{20}\), the ratio of the length of \(\overline{E F}\) to the length of \(\overline{D F}\) is \(\frac{21}{20}\). If the lengths of \(\overline{E F}\) and \(\overline{D F}\) are 21 and 20, respectively, then the ratio of the length of \(\overline{E F}\) to the length of \(\frac{20}{D F}\) is \(\frac{21}{20}\). 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 \(\overline{E F}\) to the length of \(\overline{D E}\). The length of \(\overline{D E}\) 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 a² + b² = c². Therefore, if the lengths of \(\overline{E F}\) and \(\overline{D F}\) are 21 and 20, respectively, then (21)² + (20)² = (DE)², or 841 = (DE)². Taking the positive square root of both sides of this equation yields 29 = DE. Therefore, if the lengths of \(\overline{E F}\) and \(\overline{D F}\) are 21 and 20, respectively, then the length of \(\overline{D E}\) is 29 and the ratio of the length of \(\overline{E F}\) to the length of \(\overline{D E}\) is \(\frac{21}{29}\). Thus, if \(\tan A=\frac{21}{20}\), the value of sin D is \(\frac{21}{29}\). Note that 21/29, .7241, and 0.724 are examples of ways to enter a correct answer.