Question

michele is building a dog pen as shown in the scale - drawing below. the line segments represent a fence enclosing the pen, with an opening to allow the dogs to enter and exit. each small square in the scale drawing represents a square with a side length of 3 feet. each marked point is at the vertex of a small square in the drawing. what is the measure of the angle labeled a in the scale drawing of the dog pen? a $sin^{-1}(\frac{8}{12})$ b $cos^{-1}(\frac{8}{12})$ c $\tan^{-1}(\frac{8}{12})$

Explanation:

Step1: Identify opposite and adjacent sides

Count the number of small - square side - lengths. The vertical side (opposite to angle \(A\)) has a length of 8 small - square side - lengths and the horizontal side (adjacent to angle \(A\)) has a length of 12 small - square side - lengths. Since each small - square side - length represents 3 feet, the ratio of the opposite side to the adjacent side of angle \(A\) in the right - triangle formed is based on the number of small - square side - lengths.
The ratio of the opposite side to the adjacent side of angle \(A\) is \(\frac{8}{12}\).

Step2: Recall the tangent function

The tangent of an angle in a right - triangle is defined as \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). If \(\theta = A\), then \(\tan A=\frac{8}{12}\), and \(A = \tan^{- 1}(\frac{8}{12})\).

Answer:

C. \(\tan^{-1}(\frac{8}{12})\)