Answer:
In order to construct a regular octagon inscribed in a circle within the given square, you can follow these steps:
- Begin by drawing a square using a black pencil. Label the four corners of the square as A, B, C, and D.
- Take the center of the square as point O. Use a compass to draw a circle with point O as the center and one of the corners, such as A, as a point on the circumference.
- Now, using the compass, place the needle on point A and set the width to reach point C. Draw an arc from point A that intersects the circle at point E.
- Keeping the same compass width, place the needle on point E and draw another arc that intersects the circle at point F.
- With the compass width still unchanged, place the needle on point F and draw another arc that intersects the circle at point G.
- Finally, draw a line connecting the points A, E, F, G, and C. This line will complete the octagon, with vertices at the four corners of the square.
The octagon is inscribed in the circle, which means that all eight of its vertices lie on the circle. The circle is a round shape, so all of its diameters are the same length. The perpendicular diameters that we drew in step 4 are also the same length, so the four points where the perpendicular diameters intersect the circle are the same distance from the center of the circle. This means that the four sides of the octagon that are formed by connecting these points are also the same length.
The other four sides of the octagon are formed by connecting the vertices of the square. The square is a regular square, so all of its sides are the same length. This means that the other four sides of the octagon are also the same length.
Therefore, all eight sides of the octagon are congruent.