5.4.2: Warm-up, P.99 Antonia is making four corner tables, one for each of her three sisters and herself. She has one large square piece of wood that she plans to cut into four tabletops. She begins by marking the needed cuts for the tabletops on the square piece of wood : Constructing Squares Inscribed in Circles

2 1.Each angle of the square piece of wood measures 90˚. If Antonia bisects one angle of the square, what is the measure of the two new angles? 5.4.2: Constructing Squares Inscribed in Circles 45˚(90°/2)

2.Bisect one angle of the square. Extend the angle bisector so that it intersects the square in two places. Where does the bisector intersect the square? The angle bisector intersects the square at the bisected angle and the angle opposite the bisected angle : Constructing Squares Inscribed in Circles

3.Bisect the remaining angles of the square. If Antonia cuts along each angle bisector, what figures will she have created? Antonia will have created 4 triangles : Constructing Squares Inscribed in Circles

4.What are the measures of each of the angles of the new figures? Each triangle will have two 45˚ angles and one 90˚ angle : Constructing Squares Inscribed in Circles

5.4.2: 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 : Constructing Squares Inscribed in Circles

Key Concepts Square: 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˚. 2 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 : Constructing Squares Inscribed in Circles

Key Concepts, continued : Constructing Squares Inscribed in Circles Constructing a Square Inscribed in a Circle Using a Compass 1.To construct a square inscribed in a circle, first mark the location of the center point of the circle. Label the point X. 2.Construct a circle with the sharp point of the compass on the center point. 3.Label a point on the circle point A. 4.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)

Key Concepts, continued : Constructing Squares Inscribed in Circles 5.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. 6.Extend the bisector so it intersects the circle in two places. Label the points of intersection B and D. (continued)

Key Concepts, continued : Constructing Squares Inscribed in Circles 7.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.

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 : Constructing Squares Inscribed in Circles

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

Guided Practice: Example 1, continued 1.Construct circle O : Constructing Squares Inscribed in Circles 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.

Guided Practice: Example #1, continued 2.Label a point on the circle point A : Constructing Squares Inscribed in Circles

Guided Practice: Example #1, continued 3.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 : Constructing Squares Inscribed in Circles

Guided Practice: Example #1, continued 4.Construct the perpendicular bisector of. Extend the bisector so it intersects the circle in two places. Label the points of intersection B and D : Constructing Squares Inscribed in Circles

Guided Practice: Example #1, continued 5.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 : Constructing Squares Inscribed in Circles

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

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

Guided Practice: Example #3, continued 1.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 : Constructing Squares Inscribed in Circles

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.Q : Constructing Squares Inscribed in Circles

Guided Practice: Example #3, continued 2.Label a point on the circle point J : Constructing Squares Inscribed in Circles

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

Guided Practice: Example #3, continued 4.Construct the perpendicular bisector of. Extend the bisector so it intersects the circle in two places. Label the points of intersection K and M : Constructing Squares Inscribed in Circles

Guided Practice: Example #3, continued 5.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 : Constructing Squares Inscribed in Circles

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

Homework 1)Workbook(5.4.2): P.105 # 1-10 (Use P.104, for extra space) 2) Notes(U5-134) a)Intro: read only b)Key Concepts(2): Copy the 2 charts c)?’s and summary d)Workbook: P.111, #1 27