1. Introduction
Triangles are not the only figures that can be inscribed in a circle. It is also possible to inscribe other figures, such as squares. The process for inscribing a square in a circle uses previously learned skills, including constructing perpendicular bisectors.
1.3.2: Constructing Squares Inscribed in Circles

2. Key Concepts A square is a four-sided regular polygon.
A regular polygon is a polygon that has all sides equal and all angles equal.
The measure of each of the angles of a square is 90˚.
Sides that meet at one angle to create a 90˚ angle are perpendicular.
By constructing the perpendicular bisector of a diameter of a circle, you can construct a square inscribed in a circle.
1.3.2: Constructing Squares Inscribed in Circles

3. Key Concepts, continued
Constructing a Square Inscribed in a Circle
Using a Compass
To construct a square 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.
Use a straightedge to connect point A and point X. Extend the line through the circle, creating the diameter of the circle. Label the second point of intersection C.
(continued)
1.3.2: Constructing Squares Inscribed in Circles

4. Key Concepts, continued
Construct the perpendicular bisector of by putting the sharp point of your compass on endpoint A. Open the compass wider than half the distance of Make a large arc intersecting Without changing your compass setting, put the sharp point of the compass on endpoint C. Make a second large arc. Use your straightedge to connect the points of intersection of the arcs.
Extend the bisector so it intersects the circle in two places. Label the points of intersection B and D.
(continued)
1.3.2: Constructing Squares Inscribed in Circles

5. Key Concepts, continued
Use a straightedge to connect points A and B, B and C, C and D, and A and D.
Do not erase any of your markings.
Quadrilateral ABCD is a square inscribed in circle X.
1.3.2: Constructing Squares Inscribed in Circles

6. 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
not extending segments long enough to find the vertices of the square
1.3.2: Constructing Squares Inscribed in Circles

7. Guided Practice Example 1 Construct square ABCD inscribed in circle O.
1.3.2: Constructing Squares Inscribed in Circles

8. Guided Practice: Example 1, continued Construct circle O.
Mark the location of the center point of the circle, and label the point O. Construct a circle with the sharp point of the compass on the
center point.
1.3.2: Constructing Squares Inscribed in Circles

9. Guided Practice: Example 1, continued
Label a point on the circle point A.
1.3.2: Constructing Squares Inscribed in Circles

10. Guided Practice: Example 1, continued
Construct the diameter of the circle.
Use a straightedge to
connect point A and
point O. Extend the line
through the circle,
creating the diameter
of the circle. Label the
second point
of intersection C.
1.3.2: Constructing Squares Inscribed in Circles

11. Guided Practice: Example 1, continued
Construct the perpendicular bisector of
Extend the bisector
so it intersects the
circle in two places.
Label the points of
intersection B and D.
1.3.2: Constructing Squares Inscribed in Circles

12. Guided Practice: Example 1, continued
Construct the sides of the square.
Use a straightedge to connect points A and B, B and C, C and D, and A and D, as shown on the next slide. Do not erase any of your markings.
1.3.2: Constructing Squares Inscribed in Circles

13. ✔ Guided Practice: Example 1, continued
Quadrilateral ABCD is a square inscribed in circle O. ✔
1.3.2: Constructing Squares Inscribed in Circles

14. Guided Practice: Example 1, continued
1.3.2: Constructing Squares Inscribed in Circles

15. Guided Practice Example 3
Construct square JKLM inscribed in circle Q with the radius equal to one-half the length of .
1.3.2: Constructing Squares Inscribed in Circles

16. Guided Practice: Example 3, continued Construct circle Q.
Mark the location of the center point of the circle, and label the point Q. Bisect the length of Label the midpoint of the segment as point P, as shown on the next slide.
1.3.2: Constructing Squares Inscribed in Circles

17. Guided Practice: Example 3, continued
1.3.2: Constructing Squares Inscribed in Circles

18. Guided Practice: Example 3, continued
Next, set the opening
of the compass equal
to the length of
Construct a circle
with the sharp point
of the compass
on the center point, Q.
1.3.2: Constructing Squares Inscribed in Circles

19. Guided Practice: Example 3, continued
Label a point on the circle point J.
1.3.2: Constructing Squares Inscribed in Circles

20. Guided Practice: Example 3, continued
Construct the diameter of the circle.
Use a straightedge
to connect point J
and point Q. Extend the
line through the circle,
creating the diameter
of the circle. Label the
second point of
intersection L.
1.3.2: Constructing Squares Inscribed in Circles

21. Guided Practice: Example 3, continued
Construct the perpendicular bisector of
Extend the bisector
so it intersects the
circle in two places.
Label the points
of intersection K and M.
1.3.2: Constructing Squares Inscribed in Circles

22. Guided Practice: Example 3, continued
Construct the sides of the square.
Use a straightedge to connect points J and K, K and L, L and M, and M and J, as shown on the next slide. Do not erase any of your markings.
1.3.2: Constructing Squares Inscribed in Circles

23. ✔ Guided Practice: Example 3, continued
Quadrilateral JKLM is a square inscribed in circle Q. ✔
1.3.2: Constructing Squares Inscribed in Circles

24. Guided Practice: Example 3, continued
1.3.2: Constructing Squares Inscribed in Circles