1. Day 41 – Equilateral triangle inscribed in a circle

2. Introduction
Some plane figures can drawn inside a circle in such a way that the vertices touch the edge of the circle. For instance, a triangle can be drawn inside a circle such that all its three vertices touch the edge of the circle. The common types of triangles can be drawn inside a circle, however, in this lesson, we are going to learn how to construct an equilateral triangle inside a circle.

3. Vocabulary Equilateral triangle
A triangle that has all the three sides equal and all the three angles equal.
Inscribed triangle
A triangle drawn inside another plane figure such that the vertices of the triangle touch the edge of the plane figure, in this case a circle.

4. Constructing an equilateral triangle inscribed in a circle
We are going to construct an equilateral triangle inside a circle in such a way that its vertices touch the edge of the circle using a straightedge and a compass. When a triangle is constructed inside a circle as described above, we say that the triangle has been inscribed in the circle. The method used is based on the basic idea that a turn is equivalent to 360° and an equilateral triangle has all angles measuring 60°, therefore the circle is divided into six arcs each equivalent to 60°.

5. To construct an equilateral triangle inscribed in a circle: 1
To construct an equilateral triangle inscribed in a circle: 1. We use a compass to construct a circle with center O, of any convenient radius as shown below. O

6. 2. Using a straightedge, we draw a line segment from the center to the edge of the circle as shown below. We then label the other endpoint of the segment P as shown below. OP is the radius of this circle. O P

7. 3. Using the same radius, we use point P as the center and make an arc to intersect the circle. We label this intersection point Q as shown below. O P Q

8. 4. Using the same radius, we use point Q as the center and draw an arc that intersects the circle at point R as shown below. O P Q R

9. 5. We continue to use the new intersection points constructed as the new centers with the same radius until the last arc intersects point P as shown below. We draw a total of 6 arcs. O P Q R

10. 6. We construct the sides of the equilateral triangle by selecting any intersection point and joining it to the third intersection point using a straightedge as shown below. We repeat this process until the inscribed equilateral triangle is formed as shown below. O P Q R

11. Example Draw a circle of a suitable radius using a compass and construct an equilateral triangle inscribed in the circle with the help of a straightedge.

12. Solution We draw a circle of any convenient radius and use the same radius to draw six arcs at equal intervals as shown below.

13. We join the arcs with straight lines using a straightedge in the format shown below to form the inscribed triangle.

14. homework
Use a pair of compasses and a straightedge to construct the largest possible equilateral triangle that will fit in the circle with center O shown below. O

15. Answers to homework
The largest equilateral triangle drawn is the inscribed equilateral triangle. We use the center to get the radius.

16. THE END