Question

michelle 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 will be the length, in feet, of the fence enclosing michelles dog pen?

Explanation:

Step1: Count horizontal segments

There are 3 horizontal segments. One is 2 small - square lengths, another is 1 small - square length, and the last one is 1 small - square length. In total, the number of small - square lengths for horizontal segments is \(2 + 1+1=4\). Each small - square side length represents 3 feet, so the total horizontal length is \(4\times3 = 12\) feet.

Step2: Count vertical segments

There are 2 vertical segments. One is 4 small - square lengths and the other is 1 small - square length. In total, the number of small - square lengths for vertical segments is \(4 + 1=5\). Each small - square side length represents 3 feet, so the total vertical length is \(5\times3=15\) feet.

Step3: Calculate length of diagonal segment

The diagonal segment forms the hypotenuse of a right - triangle with legs of 3 small - square lengths (vertical) and 3 small - square lengths (horizontal). The length of each small - square side is 3 feet, so the legs of the right - triangle have lengths \(3\times3 = 9\) feet each. Using the Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}\), where \(a = 9\) and \(b = 9\), we have \(c=\sqrt{9^{2}+9^{2}}=\sqrt{81 + 81}=\sqrt{162}=9\sqrt{2}\approx9\times1.414 = 12.726\) feet.

Step4: Calculate total length of fence

The total length of the fence is the sum of horizontal, vertical, and diagonal lengths. \(L=12 + 15+12.726=39.726\approx39.73\) feet.

Answer:

Approximately 39.73 feet