In a triangle ABC, the measure of angle B is 90 degrees, BC = 16 and AC = 20. Triangle DEF is similar to triangle ABC, where vertices D, E and F correspond to vertices A, B and C, respectively, and each side of triangle DEF is 1/3 the length of the corresponding side of triangle ABC. What is the value of sine F?
Answer:
Triangle ABC is a right triangle with its right angle at B. Therefore, AC is the hypotenuse of right triangle ABC, and AB and BC are the legs of right triangle ABC. According to the Pythagorean theorem,
\[ AB = \sqrt{20^2  16^2} = \sqrt{400  256} = \sqrt{144} = 12 \]
Since triangle DEF is similar to triangle ABC, with vertex F corresponding to vertex C, the measure of angle F equals the measure of angle C. Therefore, sinF = sinC. From the side lengths of triangle ABC,
\[ sinF = \frac{opposite side}{hypotenuse} = \frac{AB}{BC} = \frac{12}{20} = \frac{3}{5}\]
Therefore, sinF= 3/5 or 0.6.
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