Introduction The owners of a radio station want to build a new broadcasting building located within the triangle formed by the cities of Atlanta, Columbus,

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Presentation transcript:

Introduction The owners of a radio station want to build a new broadcasting building located within the triangle formed by the cities of Atlanta, Columbus, and Macon. Where should the station be built so that it is equidistant from each city? In this lesson, we will investigate the point that solves this problem and the geometry that supports it : Constructing Circumscribed Circles

Key Concepts To determine the location of the broadcasting building, the station owners would first need to determine the point at which the building would be equidistant from each of the three cities. To determine this point, the owners would need to find the perpendicular bisector of each of the sides of the triangle formed by the three cities. The perpendicular bisector of a segment is a line that intersects a segment at its midpoint at a right angle : Constructing Circumscribed Circles

Key Concepts, continued When all three perpendicular bisectors of a triangle are constructed, the rays intersect at one point. This point of concurrency is called the circumcenter. The circumcenter is equidistant from the three vertices of the triangle and is also the center of the circle that contains the three vertices of the triangle. A circle that contains all the vertices of a polygon is referred to as the circumscribed circle : Constructing Circumscribed Circles

Key Concepts, continued When the circumscribed circle is constructed, the triangle is referred to as an inscribed triangle, a triangle whose vertices are tangent to a circle : Constructing Circumscribed Circles

Key Concepts, continued To construct the circumscribed circle, determine the shortest distance from the circumcenter to each of the vertices of the triangle. Remember that the shortest distance from a point to a line or line segment is a perpendicular line. Follow the steps in the Guided Practice examples that follow to construct a perpendicular bisector in order to find the circumcenter of the triangle : Constructing Circumscribed Circles

Common Errors/Misconceptions confusing the construction of the incenter with the construction of the circumcenter associating the incenter (angle bisectors) with being equidistant from the vertices, and associating the circumcenter with being equidistant from the sides : Constructing Circumscribed Circles

Guided Practice Example 1 Verify that the perpendicular bisectors of acute are concurrent and that this concurrent point is equidistant from each vertex : Constructing Circumscribed Circles

Guided Practice: Example 1, continued 1.Construct the perpendicular bisector of Set the compass at point A and swing an arc. Using the same radius, swing an arc from point B so that the two arcs intersect. If they do not intersect, lengthen the radius for each arc. Connect the two intersections to locate the perpendicular bisector : Constructing Circumscribed Circles

Guided Practice: Example 1, continued : Constructing Circumscribed Circles

Guided Practice: Example 1, continued 2.Repeat the process for and : Constructing Circumscribed Circles

Guided Practice: Example 1, continued 3.Locate the point of concurrency. Label this point D. The point of concurrency is where all three perpendicular bisectors meet : Constructing Circumscribed Circles

Guided Practice: Example 1, continued 4.Verify that the point of concurrency is equidistant from each vertex. Use your compass and carefully measure the length from point D to each vertex. The measurements are the same : Constructing Circumscribed Circles ✔

Guided Practice: Example 1, continued : Constructing Circumscribed Circles

Guided Practice Example 2 Construct a circle circumscribed about acute : Constructing Circumscribed Circles

Guided Practice: Example 2, continued 1.Construct the perpendicular bisectors of each side. Use the same method as described in Example 1 to bisect each side. Label the point of concurrency of the perpendicular bisectors as point D : Constructing Circumscribed Circles

Guided Practice: Example 2, continued : Constructing Circumscribed Circles

Guided Practice: Example 2, continued 2.Verify that the point of concurrency is equidistant from each of the vertices. Use your compass and carefully measure the length of each perpendicular from D to each vertex. Verify that the measurements are the same : Constructing Circumscribed Circles

Guided Practice: Example 2, continued 3.Construct the circumscribed circle with center D. Place the compass point at D and open the compass to the distance of D to any vertex. Use this setting to construct a circle around : Constructing Circumscribed Circles

Guided Practice: Example 2, continued : Constructing Circumscribed Circles

Guided Practice: Example 2, continued Circle D is circumscribed about and each vertex is tangent to the circle : Constructing Circumscribed Circles ✔

Guided Practice: Example 2, continued : Constructing Circumscribed Circles