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Geometry and Trigonometry Difficulty: Hard

In triangle $X Y Z$ , angle $Z$ is a right angle and the length of $\bar{Y Z}$ is $24$ units. If $\tan X = \frac{12}{35}$ , what is the perimeter, in units, of triangle $X Y Z$ ?

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Explanation

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 $\tan X = \frac{Y Z}{X Z}$ . It's given that $\tan X = \frac{12}{35}$ and the length of side $Y Z$ is $24$ units. Substituting $StartFraction 12 Over 35 EndFraction$ for $\tan X$ and $24$ for $Y Z$ in the equation $\tan X = \frac{Y Z}{X Z}$ yields $\frac{12}{35} = \frac{24}{X Z}$ . Multiplying both sides of this equation by $35 left parenthesis upper X upper Z right parenthesis$ yields $12 left parenthesis upper X upper Z right parenthesis equals 24 left parenthesis 35 right parenthesis$ , or $12 left parenthesis upper X upper Z right parenthesis equals 840$ . Dividing both sides of this equation by $12$ yields $upper X upper Z equals 70$ . 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 $5,476 equals c squared$ . Taking the square root of both sides of this equation yields $\pm 74 = c$ . Since the length of the hypotenuse must be positive, $74 equals c$ . 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.