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Problem 620

# Inscribe a square in a given semicircle.

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To inscribe a square in a given semicircle, first draw the semicircle and diameter AB with center O. Mark the midpoint C of diameter AB, construct two perpendicular bisectors to AB at points A and C, meeting at point P. Then, construct a perpendicular bisector to segment PB at point B, intersecting the semicircle at point Q. The vertices of the inscribed square are A, P, B, and Q, with side length s, where $$2s = d$$ (d is the diameter of the semicircle).
See the step by step solution

## Step 1: Draw the semicircle and diameter

Draw a semicircle with center O and diameter AB. The diameter AB will act as the base of the inscribed square and will further help in positioning the vertices of the square.

## Step 2: Mark the midpoint of the diameter

Mark the midpoint C of diameter AB. Since the square is inscribed in the semicircle, points A and B will be the opposite vertices of the square, and its side length will be half the diameter. So, AC = CB = s, where s is the side length of the square.

## Step 3: Construct perpendicular bisectors to diameter AB

Construct two perpendicular bisectors to diameter AB at points A and C. They will intersect at point P, which will be the third vertex of the inscribed square. Angle ACB will now be 90 degrees since A, C, and B are part of the inscribed square.

## Step 4: Find the fourth vertex by constructing a perpendicular bisector

Construct a perpendicular bisector to segment PB at point B, which will intersect the semicircle at point Q, which will be the fourth vertex of the inscribed square.

## Step 5: Determine the side length of the square

Since we now have all four vertices of the inscribed square (A, P, B, Q), we can determine the side length of the square by finding the length of segment AC, CB, or any other side. Each side length will be half the diameter of the semicircle. The inscribed square in the given semicircle is now successfully constructed with vertices A, P, B, and Q. The relationship between the side length of the square and the diameter of the semicircle is 2s = d, where s is the side length of the square and d is the diameter of the semicircle.

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