In triangle X Y Z , angle Y is a right angle, the measure of angle Z is 33 °

Question

In triangle X Y Z , angle Y is a right angle, the measure of angle Z is 33 &#176; , and the length of Y Z &#175; is 26 units. If the area, in square units, of triangle X Y Z can be represented by the expression k tan 33 &#176; , where k is a constant, what is the value of k ?

Accepted Answer

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, X Y &macr; is the leg opposite of angle Z and Y Z &macr; is the leg adjacent to angle Z . It follows that tan Z = X Y Y Z . It's also given that the measure of angle Z is 33 &deg; and the length of Y Z &macr; is 26 units. Substituting 33 &deg; for Z and 26 for Y Z in the equation tan Z = X Y Y Z yields tan 33 &deg; = X Y 26 . Multiplying each side of this equation by 26 yields 26 tan 33 &deg; = X Y . Therefore, the length of X Y &macr; is 26 tan 33 &deg; . The area of a triangle is half the product of the lengths of its legs. Since the length of Y Z &macr; is 26 and the length of X Y &macr; is 26 tan 33 &deg; , it follows that the area of triangle X Y Z is 1 2 26 26 tan 33 &deg; square units, or 338 tan 33 &deg; square units. It's given that the area, in square units, of triangle X Y Z can be represented by the expression k tan 33 &deg; , where k is a constant. Therefore, 338 is the value of k .

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