Problem 47 - Precalculus: A Concise Course

Question

Find the length of the sides of a regular pentagon inscribed in a circle of radius 25 inches.

Step-by-Step Solution

Answer

The length of each side of the regular pentagon is \(50 * sin(36)\) inches

Step 1: Draw a sketch

Start by drawing a regular pentagon inscribed in a circle. For convenience, you can choose one vertex of pentagon as the center of circle. Now, choose one side of pentagon and draw two radii from its endpoints to the center of the circle. This splits the side of the pentagon into two congruent segments and forms an isosceles triangle with the side of the pentagon.

Step 2: Calculate the central angle of each triangle

A regular pentagon divides a circle into five equal parts. Therefore, the central angle for each of these parts (or triangles) is \(\frac{360}{5} = 72\) degrees.

Step 3: Find the length of a side

Each triangle is now an isosceles triangle. This means the angle opposite each side of the pentagon (let's call this \(s\)) is \(\frac{72}{2} = 36\) degrees. Using this, we can derive expression for \(s\) from the trigonometric definition of sine which states that the sine of an angle in a right triangle is the length of the opposite side divided by the length of the hypotenuse; so \(sin(36) = \frac{s/2}{25}\). Solving this equation for \(s\) gives \(s = 2 * 25 * sin(36)\)

Step 4: Calculate the length

Finally, substitute the value of sine 36 degrees and then simplify to get the length of the side.