UNIT 1 SIMILARITY, CONGRUENCE, AND PROOFS Lesson 3: Constructing Polygons Instruction

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1 rerequisite Skills This lesson requires the use of the following skills: using a compass copying and bisecting line segments constructing perpendicular lines constructing circles of a given radius 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. 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. U1-164

2 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. 5. Construct the perpendicular bisector of AC by putting the sharp point of your compass on endpoint A. Open the compass wider than half the distance of AC. Make a large arc intersecting AC. 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. 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 U1-165

3 Guided ractice Example 1 Construct square ABCD inscribed in circle O. 1. 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. O 2. Label a point on the circle point A. A O U1-166

4 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. A O C 4. Construct the perpendicular bisector of AC. Extend the bisector so it intersects the circle in two places. Label the points of intersection B and D. A B O D C U1-167

5 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. Do not erase any of your markings. B A O D C Quadrilateral ABCD is a square inscribed in circle O. U1-168

6 Example 2 Construct square EFGH inscribed in circle with the radius equal to the length of E. E 1. Construct circle. Mark the location of the center point of the circle, and label the point. Set the opening of the compass equal to the length of E. Construct a circle with the sharp point of the compass on the center point,. 2. Label a point on the circle point E. E U1-169

7 3. Construct the diameter of the circle. Use a straightedge to connect point E and point. Extend the line through the circle, creating the diameter of the circle. Label the second point of intersection G. G E 4. Construct the perpendicular bisector of EG. Extend the bisector so it intersects the circle in two places. Label the points of intersection F and H. G F H E U1-170

8 5. Construct the sides of the square. Use a straightedge to connect points E and F, F and G, G and H, and H and E. Do not erase any of your markings. G F H E Quadrilateral EFGH is a square inscribed in circle. U1-171

9 Example 3 Construct square KLM inscribed in circle Q with the radius equal to one-half the length of L. L 1. Construct circle Q. Mark the location of the center point of the circle, and label the point Q. Bisect the length of L. Label the midpoint of the segment as point. L 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 U1-172

10 2. Label a point on the circle point. Q 3. Construct the diameter of the circle. Use a straightedge to connect point and point Q. Extend the line through the circle, creating the diameter of the circle. Label the second point of intersection L. L Q U1-173

11 4. Construct the perpendicular bisector of L. Extend the bisector so it intersects the circle in two places. Label the points of intersection K and M. L M Q K 5. Construct the sides of the square. Use a straightedge to connect points and K, K and L, L and M, and M and. Do not erase any of your markings. L M Q K Quadrilateral KLM is a square inscribed in circle Q. U1-174

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