1. Day 44 – Summary of inscribed figures

2. Introduction
If a geometric figure is constructed inside a circle such that its vertices are on the circumference of the circle, such a figure is said to be inscribed in the circle. In this lesson, we are going to discuss a summary of how an equilateral triangle, a square, and a regular hexagon can be inscribed inside a circle.

3. Vocabulary Regular hexagon
This is a six sided figure with each side equal to the distance from its center to one of its vertices.

4. Construction of an equilateral triangle in a circle
To construct an equilateral triangle in a circle, we follow the steps given below.
Draw a circle of any convenient radius.
Position the needle of the compass at the center of the circle and extend the compass to the circumference of the circle.
Keeping the same compass width, position the compass on the circumference of the circle and make an arc that intersects the circle.

5. 4. Next, position the compass on the intersection the arc and draw another arc on the circumference a shown. 5. Keeping the same compass width, continue this way until you have six arcs on the circumference of the circle.

6. 6. We construct the sides of the equilateral triangle by selecting any intersection point and joining it to the arc after the adjacent arc from any given arc using a straightedge or a ruler as shown below. We repeat this process until the inscribed equilateral triangle is formed as shown below.

7. Construction of a regular hexagon in a circle
A regular hexagon has its sides equal to the distance from its center to one of its vertices.
The following steps are followed to construct a regular hexagon inscribed in a circle.
Draw a circle of any convenient radius.
Position the needle of the compass at the center of the circle and extend the compass to the circumference of the circle.

8. 3. Keeping the same compass width, position the compass on the circumference of the circle and make an arc that intersects the circle. 4. Position the compass on the intersection the arc and draw another arc on the circumference a shown.

9. 5. Keeping the same compass width, continue this way until you have six arcs on the circumference of the circle.

10. 6. We construct the sides of the regular hexagon by selecting any intersection point and joining it to the intermediate intersection point using a straightedge as shown below. We repeat this process until the inscribed regular hexagon is formed as shown below.

11. Construction of a square in a circle Construction of a square in a circle is based on the idea that the diagonals of a square meet at a right angle. Thus if the diameter of a circle is one of the diagonals, a perpendicular bisector to the diameter will be the other diagonal. To construct a square in a circle we follow the steps below. 1. Draw a circle of any convenient diameter as shown.

12. 2. Construct a perpendicular bisector to the diameter as shown below. 3. Join the points at which the diameter intersects the circumference and points at which the perpendicular bisector intersects the circle as shown.

13. Example Construct an inscribed square in the circle given below
Example Construct an inscribed square in the circle given below. Solution

14. homework
Construct an inscribed regular hexagon in a circle with a diameter of 2 inches.

15. Answers to homework .

16. THE END