1. Introduction
The ability to copy and bisect angles and segments, as well as construct perpendicular and parallel lines, allows you to construct a variety of geometric figures, including triangles, squares, and hexagons. There are many ways to construct these figures and others. Sometimes the best way to learn how to construct a figure is to try on your own. You will likely discover different ways to construct the same figure and a way that is easiest for you. In this lesson, you will learn two methods for constructing a triangle within a circle.
1.3.1: Constructing Equilateral Triangles Inscribed in Circles

2. Key Concepts Triangles
A triangle is a polygon with three sides and three angles.
There are many types of triangles that can be constructed.
Triangles are classified based on their angle measure and the measure of their sides.
Equilateral triangles are triangles with all three sides equal in length.
The measure of each angle of an equilateral triangle is 60˚.
1.3.1: Constructing Equilateral Triangles Inscribed in Circles

3. Key Concepts, continued
Circles
A circle is the set of all points that are equidistant from a reference point, the center.
The set of points forms a two-dimensional curve
that is 360˚.
Circles are named by their center. For example, if a circle has a center point, G, the circle is named circle G.
1.3.1: Constructing Equilateral Triangles Inscribed in Circles

4. Key Concepts, continued
The diameter of a circle is a straight line that goes through the center of a circle and connects two points on the circle. It is twice the radius.
The radius of a circle is a line segment that runs from the center of a circle to a point on the circle.
The radius of a circle is one-half the length of the diameter.
There are 360˚ in every circle.
1.3.1: Constructing Equilateral Triangles Inscribed in Circles

5. Key Concepts, continued
Inscribing Figures
To inscribe means to draw a figure within another figure so that every vertex of the enclosed figure touches the outside figure.
A figure inscribed within a circle is a figure drawn within a circle so that every vertex of the figure touches the circle.
1.3.1: Constructing Equilateral Triangles Inscribed in Circles

6. Key Concepts, continued
It is possible to inscribe a triangle within a circle. Like with all constructions, the only tools used to inscribe a figure are a straightedge and a compass, patty paper and a straightedge, reflective tools and a straightedge, or technology.
This lesson will focus on constructions with a compass and a straightedge.
1.3.1: Constructing Equilateral Triangles Inscribed in Circles

7. Key Concepts, continued
Method 1: Constructing an Equilateral Triangle Inscribed in a Circle Using a Compass
To construct an equilateral triangle inscribed in a circle, first mark the location of the center point of the circle. Label the point X.
Construct a circle with the sharp point of the compass on the center point.
Label a point on the circle point A.
Without changing the compass setting, put the sharp point of the compass on A and draw an arc to intersect the circle at two points. Label the points B and C.
Use a straightedge to construct
(continued)
1.3.1: Constructing Equilateral Triangles Inscribed in Circles

8. Key Concepts, continued
Put the sharp point of the compass on point B. Open the compass until it extends to the length of Draw another arc that intersects the circle. Label the point D.
Use a straightedge to construct and
Do not erase any of your markings.
Triangle BCD is an equilateral triangle inscribed in circle X.
1.3.1: Constructing Equilateral Triangles Inscribed in Circles

9. Key Concepts, continued
A second method “steps out” each of the vertices.
Once a circle is constructed, it is possible to divide the circle into 6 equal parts.
Do this by choosing a starting point on the circle and moving the compass around the circle, making marks that are the length of the radius apart from one another.
Connecting every other point of intersection results in an equilateral triangle.
1.3.1: Constructing Equilateral Triangles Inscribed in Circles

10. Key Concepts, continued
Method 2: Constructing an Equilateral Triangle Inscribed in a Circle Using a Compass
To construct an equilateral triangle inscribed in a circle, first mark the location of the center point of the circle. Label the point X.
Construct a circle with the sharp point of the compass on the center point.
Label a point on the circle point A.
Without changing the compass setting, put the sharp point of the compass on A and draw an arc to intersect the circle at one point. Label the point of intersection B.
(continued)
1.3.1: Constructing Equilateral Triangles Inscribed in Circles

11. Key Concepts, continued
Put the sharp point of the compass on point B and draw an arc to intersect the circle at one point. Label the point of intersection C.
Continue around the circle, labeling points D, E, and F. Be sure not to change the compass setting.
Use a straightedge to connect A and C, C and E, and E and A.
Do not erase any of your markings.
Triangle ACE is an equilateral triangle inscribed in circle X.
1.3.1: Constructing Equilateral Triangles Inscribed in Circles

12. Common Errors/Misconceptions
inappropriately changing the compass setting
attempting to measure lengths and angles with rulers and protractors
not creating large enough arcs to find the points of intersection
1.3.1: Constructing Equilateral Triangles Inscribed in Circles

13. Guided Practice Example 3
Construct equilateral triangle JKL inscribed in circle P using Method 1. Use the length of as the radius for circle P.
1.3.1: Constructing Equilateral Triangles Inscribed in Circles

14. Guided Practice: Example 3, continued Construct circle P.
Mark the location of the center point of the circle, and label the point P. Set the opening of the compass equal to the length of Then, put the sharp point of the compass on point P and construct a circle. Label a point on the circle point G.
1.3.1: Constructing Equilateral Triangles Inscribed in Circles

15. Guided Practice: Example 3, continued
Locate vertices J and K of the equilateral triangle. Without changing the compass setting, put the sharp point of the compass on G. Draw an arc to intersect the circle at two points. Label the points J and K, as shown on the next slide.
1.3.1: Constructing Equilateral Triangles Inscribed in Circles

16. Guided Practice: Example 3, continued
1.3.1: Constructing Equilateral Triangles Inscribed in Circles

17. Guided Practice: Example 3, continued
Locate the third vertex of the equilateral triangle.
Put the sharp point of the compass on point J. Open the compass until it extends to the length of Draw another arc that intersects the circle, and label the point L, as shown on the next slide.
1.3.1: Constructing Equilateral Triangles Inscribed in Circles

18. Guided Practice: Example 3, continued
1.3.1: Constructing Equilateral Triangles Inscribed in Circles

19. Guided Practice: Example 3, continued
Construct the sides of the triangle.
Use a straightedge to connect J and K, K and L, and L and J, as shown on the next slide. Do not erase any of your markings.
1.3.1: Constructing Equilateral Triangles Inscribed in Circles

20. ✔ Guided Practice: Example 3, continued
Triangle JKL is an equilateral triangle inscribed in circle P with the given radius. ✔
1.3.1: Constructing Equilateral Triangles Inscribed in Circles

21. Guided Practice: Example 3, continued
1.3.1: Constructing Equilateral Triangles Inscribed in Circles

22. Guided Practice Example 4
Construct equilateral triangle JLN inscribed in circle P using Method 2. Use the length of as the radius for circle P.
1.3.1: Constructing Equilateral Triangles Inscribed in Circles

23. Guided Practice: Example 4, continued Construct circle P.
Mark the location of the center point of the circle, and label the point P. Set the opening of the compass equal to the length of Then, put the sharp point of the compass on point P and construct a circle. Label a point on the circle point G.
1.3.1: Constructing Equilateral Triangles Inscribed in Circles

24. Guided Practice: Example 4, continued Locate vertex J.
Without changing the
compass setting, put the
sharp point of the
compass on G. Draw
an arc to intersect
the circle at one point.
Label the point of
intersection J.
1.3.1: Constructing Equilateral Triangles Inscribed in Circles

25. Guided Practice: Example 4, continued
Put the sharp point of the compass on point J. Without changing the compass setting, draw an arc to intersect the circle at one point. Label the point of intersection K.
Continue around the circle, labeling points L, M, and N, as shown on the next slide. Be sure not to change the compass setting.
1.3.1: Constructing Equilateral Triangles Inscribed in Circles

26. Guided Practice: Example 4, continued
1.3.1: Constructing Equilateral Triangles Inscribed in Circles

27. Guided Practice: Example 4, continued
Construct the sides of the triangle.
Use a straightedge to connect J and L, L and N, and J and N, as shown on the next slide. Do not erase any of your markings.
1.3.1: Constructing Equilateral Triangles Inscribed in Circles

28. ✔ Guided Practice: Example 4, continued
Triangle JLN is an equilateral triangle inscribed in circle P. ✔
1.3.1: Constructing Equilateral Triangles Inscribed in Circles

29. Guided Practice: Example 4, continued
1.3.1: Constructing Equilateral Triangles Inscribed in Circles