In triangle X Y Z , angle Z is a right angle and the length of Y Z ¯ is

Question

In triangle X Y Z , angle Z is a right angle and the length of Y Z &#175; is 24 units. If tan X = 12 35 , what is the perimeter, in units, of triangle X Y Z ?

Accepted Answer

Choice B is correct. It's given that angle Z in triangle X Y Z is a right angle. Thus, side Y Z is the leg opposite angle X and side X Z is the leg adjacent to angle X . The tangent of an acute angle in a right triangle is the ratio of the length of the leg opposite the angle to the length of the leg adjacent to the angle. It follows that tan X = Y Z X Z . It's given that tan X = 12 35 and the length of side Y Z is 24 units. Substituting 12 35 for tan X and 24 for Y Z in the equation tan X = Y Z X Z yields 12 35 = 24 X Z . Multiplying both sides of this equation by 35 X Z yields 12 X Z = 24 35 , or 12 X Z = 840 . Dividing both sides of this equation by 12 yields X Z = 70 . The length X Y can be calculated using the Pythagorean theorem, which states that if a right triangle has legs with lengths of a and b and a hypotenuse with length c , then a 2 + b 2 = c 2 . Substituting 70 for a and 24 for b in this equation yields 70 2 + 24 2 = c 2 , or 5,476 = c 2 . Taking the square root of both sides of this equation yields &plusmn; 74 = c . Since the length of the hypotenuse must be positive, 74 = c . Therefore, the length of X Y is 74 units. The perimeter of a triangle is the sum of the lengths of all sides. Thus, 74 + 70 + 24 units, or 168 units, is the perimeter of triangle X Y Z . Choice A is incorrect and may result from conceptual or calculation errors. Choice C is incorrect. This would be the perimeter, in units, for a right triangle where the length of side Y Z is 12 units, not 24 units. Choice D is incorrect and may result from conceptual or calculation errors.

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