1. **Problem statement:** Explain how the angle in a semicircle theorem enables construction of a right-angled triangle with a given hypotenuse AB. 2. **Theorem used:** The angle in a semicircle theorem states that any triangle inscribed in a circle where one side is the diameter is a right triangle, with the right angle opposite the diameter. 3. **Explanation:** Given segment AB as the hypotenuse, draw a circle with AB as diameter. 4. Any point C on the circle (other than A and B) forms triangle ABC with right angle at C. 5. This constructs a right-angled triangle with hypotenuse AB. 1. **Problem statement:** Construct a square equal in area to a given rectangle. 2. **Formula:** Area of rectangle = length \( l \times \) width \( w \). 3. To construct a square with the same area, side \( s \) must satisfy \( s^2 = l \times w \). 4. Use geometric mean construction: Draw segment of length \( l + w \). 5. Mark points A and B such that AB = \( l + w \). 6. Find midpoint M of AB and draw semicircle with diameter AB. 7. Erect perpendicular at point dividing AB into segments \( l \) and \( w \) to intersect semicircle at point C. 8. Then AC is the side of the square with area equal to the rectangle. 1. **Problem statement:** Construct a right triangle equal in area to a given triangle. 2. **Formula:** Area of triangle = \( \frac{1}{2} \times base \times height \). 3. Given triangle with area \( A \), construct a right triangle with legs \( x \) and \( y \) such that \( \frac{1}{2}xy = A \). 4. Choose one leg length arbitrarily, then calculate the other leg as \( y = \frac{2A}{x} \). 5. Construct the right triangle with these legs. 1. **Problem statement:** Construct a square equal in area to a given triangle. 2. **Formula:** Area of triangle = \( A \). 3. Side of square \( s \) satisfies \( s^2 = A \). 4. Construct right triangle equal in area to given triangle (from previous step). 5. Then construct square equal in area to that right triangle (from rectangle to square method). 1. **Problem statement:** Construct a square equal in area to the sum of two given squares. 2. **Formula:** If squares have sides \( a \) and \( b \), sum of areas = \( a^2 + b^2 \). 3. By Pythagorean theorem, construct right triangle with legs \( a \) and \( b \). 4. Hypotenuse \( c = \sqrt{a^2 + b^2} \) is side of square equal to sum of areas. 1. **Problem statement:** Deduce that any polygon may be squared. 2. Since any polygon can be divided into triangles, and each triangle can be squared, 3. Sum of areas of these squares can be combined using the previous construction to form a single square equal in area to the polygon. **Final answer:** Using the angle in a semicircle theorem and geometric mean constructions, one can construct right triangles and squares equal in area to given shapes, and by combining these, any polygon can be squared.