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Geometry and Trigonometry / Right triangles and trigonometry Difficulty: Hard

In triangle $X Y Z$ , angle $Y$ is a right angle, the measure of angle $Z$ is $33 degree$ , and the length of $\bar{Y Z}$ is $26$ units. If the area, in square units, of triangle $X Y Z$ can be represented by the expression $k tangent 33 degree$ , where $k$ is a constant, what is the value of $k$ ?

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Explanation

The correct answer is $338$ . 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. In triangle $X Y Z$ , it's given that angle $Y$ is a right angle. Thus, $\bar{X Y}$ is the leg opposite of angle $Z$ and $\bar{Y Z}$ is the leg adjacent to angle $Z$ . It follows that $\tan Z = \frac{X Y}{Y Z}$ . It's also given that the measure of angle $Z$ is $33 degree$ and the length of $\bar{Y Z}$ is $26$ units. Substituting $33 degree$ for $Z$ and $26$ for $Y Z$ in the equation $\tan Z = \frac{X Y}{Y Z}$ yields $\tan 33 ° = \frac{X Y}{26}$ . Multiplying each side of this equation by $26$ yields $26 tangent 33 degree equals upper X upper Y$ . Therefore, the length of $\bar{X Y}$ is $26 tangent 33 degree$ . The area of a triangle is half the product of the lengths of its legs. Since the length of $\bar{Y Z}$ is $26$ and the length of $\bar{X Y}$ is $26 tangent 33 degree$ , it follows that the area of triangle $X Y Z$ is $one half left parenthesis 26 right parenthesis left parenthesis 26 tangent 33 degree right parenthesis$ square units, or $338 tangent 33 degree$ square units. It's given that the area, in square units, of triangle $X Y Z$ can be represented by the expression $k tangent 33 degree$ , where $k$ is a constant. Therefore, $338$ is the value of $k$ .