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FeatureLesson Geometry Lesson Main (For help, go to Lesson 1-7.) Lesson 5-3 1. an angle bisector 2. a perpendicular bisector of a side 3. Draw GH Construct CD GH at the midpoint of GH. 4. Draw AB with a point E not on AB. Construct EF AB. Concurrent Lines, Medians, and Altitudes Draw a large triangle. Construct each figure. Check Skills Youll Need 5-3

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FeatureLesson Geometry Lesson Main 1–2. 3. 4. Solutions Lesson 5-3 Concurrent Lines, Medians, and Altitudes Answers may vary. Samples given: Check Skills Youll Need 5-3

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FeatureLesson Geometry Lesson Main Lesson 5-3 Concurrent Lines, Medians, and Altitudes 5-3 Warm Up 1. JK is perpendicular to ML at its midpoint K. List the congruent segments. Find the midpoint of the segment with the given endpoints. 2. (–1, 6) and (3, 0) 3. (–7, 2) and (–3, –8) (–5, –3) (1, 3)

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FeatureLesson Geometry Lesson Main

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FeatureLesson Geometry Lesson Main

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FeatureLesson Geometry Lesson Main

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FeatureLesson Geometry Lesson Main

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FeatureLesson Geometry Lesson Main Lesson 5-3 Concurrent Lines, Medians, and Altitudes 5-3 When three or more lines intersect at one point, the lines are said to be concurrent. The point of concurrency is the point where they intersect.

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FeatureLesson Geometry Lesson Main Lesson 5-3 Concurrent Lines, Medians, and Altitudes 5-3 The circumcenter can be inside the triangle, outside the triangle, or on the triangle. The point of concurrency of the three perpendicular bisectors of a triangle is the circumcenter of the triangle.

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FeatureLesson Geometry Lesson Main Lesson 5-3 Concurrent Lines, Medians, and Altitudes 5-3 The circumcenter of ΔABC is the center of its circumscribed circle. A circle that contains all the vertices of a polygon is circumscribed about the polygon.

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FeatureLesson Geometry Lesson Main Lesson 5-3 Concurrent Lines, Medians, and Altitudes 5-3 A triangle has three angles, so it has three angle bisectors. The angle bisectors of a triangle are also concurrent. This point of concurrency is the incenter of the triangle. Unlike the circumcenter, the incenter is always inside the triangle.

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FeatureLesson Geometry Lesson Main Lesson 5-3 Concurrent Lines, Medians, and Altitudes 5-3 The incenter is the center of the triangles inscribed circle. A circle inscribed in a polygon intersects each line that contains a side of the polygon at exactly one point.

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FeatureLesson Geometry Lesson Main Lesson 5-3 Concurrent Lines, Medians, and Altitudes 5-3 Circumcenter Theorem The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices. Incenter Theorem The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides.

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FeatureLesson Geometry Lesson Main Find the center of the circle that circumscribes XYZ. Because X has coordinates (1, 1) and Z has coordinates (5, 1), XZ lies on the horizontal line y = 1. The perpendicular bisector of XZ is the vertical line that passes through (, 1) or (3, 1), so the equation of the perpendicular bisector of XZ is x = 3. 1 + 5 2 Because X has coordinates (1, 1) and Y has coordinates (1, 7), XY lies on the vertical line x = 1. The perpendicular bisector of XY is the horizontal line that passes through (1, ) or (1, 4), so the equation of the perpendicular bisector of XY is y = 4. 1 + 7 2 Lesson 5-3 Concurrent Lines, Medians, and Altitudes Additional Examples 5-3 Finding the Circumcenter You need to determine the equation of two bisectors, then determine the point of intersection.

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FeatureLesson Geometry Lesson Main (continued) The lines y = 4 and x = 3 intersect at the point (3, 4). This point is the center of the circle that circumscribes XYZ. Lesson 5-3 Concurrent Lines, Medians, and Altitudes Quick Check Additional Examples 5-3

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FeatureLesson Geometry Lesson Main City planners want to locate a fountain equidistant from three straight roads that enclose a park. Explain how they can find the location. Theorem 5-7 states that the bisectors of the angles of a triangle are concurrent at a point equidistant from the sides. The city planners should find the point of concurrency of the angle bisectors of the triangle formed by the three roads and locate the fountain there. The roads form a triangle around the park. Lesson 5-3 Concurrent Lines, Medians, and Altitudes Additional Examples 5-3 Real-World Connection Quick Check

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FeatureLesson Geometry Lesson Main The point of concurrency of the perpendicular bisectors of the sides of a triangle. Circumcenter

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FeatureLesson Geometry Lesson Main Circumcenter The circumcenter is equidistant from each vertex of the triangle.

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FeatureLesson Geometry Lesson Main The point of concurrency of the three angles bisectors of the triangle. Incenter

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FeatureLesson Geometry Lesson Main The incenter is equidistant from the sides of a triangle. Incenter

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FeatureLesson Geometry Lesson Main The incenter is equidistant from the sides of a triangle. Incenter

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