OG详解-OG7 数学2 Q24

Choice B is correct. It’s given that angle Z in triangle XYZ is a right angle. Thus, side YZ is the leg opposite angle X and side XZ 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 = YZXZ​​. It’s given that tan X = 1235​​  and the length of side YZ is 24 units. Substituting 1235​​ for tan X and 24 for YZ in the equation tan X = YZXZ​​  yields  1235​=24XZ​​ . Multiplying both sides of this equation by 35(XZ) yields 12(XZ) = 24(35), or 12(XZ) = 840. Dividing both sides of this equation by 12 yields XZ = 70. The length XY 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 a² +b² = c². Substituting 70 for a and 24 for b in this equation yields 70² +24² = c², or 5,476 = c². Taking the square root of both sides of this equation yields ±74 = c. Since the length of the hypotenuse must be positive, 74 = c. Therefore, the length of XY 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 XYZ.
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 YZ is 12 units, not 24 units. Choice D is incorrect and may result from conceptual or calculation errors.