19 In triangle QRS shown, < RS. Which expression represents the length of QR/QS?
Correct answer: D (parsed from the explanation — this source omits an answer key for this item)
Explanation
Choice D is correct. The figure shows that triangle QRS is a right triangle. Each of sin cos the given choices is an expression containing Q or Q. For an acute angle in a right triangle, the sine of the angle is the length of the opposite leg divided by the length of the hypotenuse, and the cosine of the angle is the length of the sin =RS adjacent leg divided by the length of the hypotenuse. It follows that Q cos =QR < QS and Q , where QR RS. It’s given in the figure that the length of side RS 18 QS sin =RS is . Since only the equation Q contains RS, an expression representing 18 QS QS can be found by substituting for RS in this equation, which yields 18 sin = $sin =18 Q . Multiplying both sides of this equation by QS yields QS Q . QS sin = 18 Dividing both sides of this equation by Q yields QS sin . Therefore, the 18 Q expression sin represents the length of QS. Q 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 and may result from conceptual or calculation errors.
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