Question

15 mark for review in triangle xyz, angle y is a right angle, the measure of angle z is 33°, and the length of \\(\overline{yz}\\) is 26 units. if the area, in square units, of triangle xyz can be represented by the expression \\(k \tan 33^\circ\\), where \\(k\\) is a constant, what is the value of \\(k\\)? answer preview:

Explanation:

Step1: Identify triangle sides

In right triangle \( XYZ \) (right - angled at \( Y \)), \( \angle Z = 33^{\circ} \), \( YZ = 26 \) (adjacent to \( \angle Z \)). Let \( XY = x \) (opposite to \( \angle Z \)). Using the tangent function: \( \tan Z=\frac{XY}{YZ} \), so \( \tan33^{\circ}=\frac{x}{26} \), then \( x = 26\tan33^{\circ} \).

Step2: Calculate area of triangle

The area of a right - triangle is \( A=\frac{1}{2}\times\text{base}\times\text{height} \). Here, base \( YZ = 26 \) and height \( XY=x = 26\tan33^{\circ} \). So \( A=\frac{1}{2}\times26\times26\tan33^{\circ}=\frac{26\times26}{2}\tan33^{\circ}=338\tan33^{\circ} \).

Since the area is given as \( k\tan33^{\circ} \), comparing with \( A = k\tan33^{\circ} \), we get \( k = 338 \).

Answer:

\( 338 \)