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 AB is 2.9 times the length of DE. If tanA=2120, what is the value of sin D ?

Answer (Detailed Solution Below) 7241

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The correct answer is 2129. 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 tanA=2120, then tanD=2120. 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 tanD=2120, the ratio of the length of EF to the length of DF is 2120. If the lengths of EF and DF are 21 and 20, respectively, then the ratio of the length of EF to the length of 20DF is 2120. 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 EF to the length of DE. The length of DE 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+ b= c2. Therefore, if the lengths of EF and DF are 21 and 20, respectively, then (21)+ (20)= (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 EF and DF are 21 and 20, respectively, then the length of DE is 29 and the ratio of the length of EF to the length of DE is 2129. Thus, if tanA=2120, the value of sin D is 2129. Note that 21/29, .7241, and 0.724 are examples of ways to enter a correct answer.

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