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Problem 622

# Construct a regular decagon.

Expert verified
Construct a circle with a chosen radius. Draw its horizontal diameter and find its midpoint. Construct a perpendicular line from the midpoint to the circle's top-most point and calculate the decagon's side length using the golden ratio (s = r * (1 + $$\phi$$)/2). Construct the regular decagon by finding the points on the circle, equal to the calculated side length, and join the vertices with straight lines.
See the step by step solution

## Step 1: Construct a circle

Using a compass, draw a circle with a chosen radius. The circumference of this circle will be the perimeter of the decagon.

## Step 2: Draw the horizontal diameter and find the midpoint

Draw a horizontal line across the circle, creating a diameter. Find the midpoint of the diameter and label it as point M.

## Step 3: Construct a perpendicular from the midpoint to the circle's top-most point

Using the compass and straightedge, construct a perpendicular line from point M to the circle's top-most point to form an equilateral triangle. Let's call the top-most point of the circle vertex A.

## Step 4: Calculate the length of the decagon's side using the golden ratio

The golden ratio, ($$\phi$$), is approximately equal to 1.618. In a regular decagon, the side length (s) is related to the radius (r) as: s = r * (1 + $$\phi$$)/2 Calculate the side length (s) using the circle's radius (r) that you have chosen.

## Step 5: Construct the regular decagon by finding the points on the circle equidistant from each other

Starting from vertex A, draw an arc with a compass, setting its length equal to the calculated side length (s). Mark the point where the arc intersects the circle as vertex B. Repeat this process, always starting from the last marked vertex, until you have marked all ten vertices. Now, join these vertices with straight lines using your straightedge, completing the construction of a regular decagon.

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