Concurrent Lines, Medians, and Altitudes Lesson 5-3 Check Skills You’ll Need (For help, go to Lesson 1-7.) Draw a large triangle. Construct each figure. 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. Check Skills You’ll Need 5-3

Concurrent Lines, Medians, and Altitudes Lesson 5-3 Check Skills You’ll Need Solutions Answers may vary. Samples given: 1–2. 3. 4. 5-3

Concurrent Lines, Medians, and Altitudes Lesson 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) (1, 3) (–5, –3) 5-3

Concurrent Lines, Medians, and Altitudes Lesson 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. 5-3

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

Concurrent Lines, Medians, and Altitudes Lesson 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. 5-3

Unlike the circumcenter, the incenter is always inside the triangle. Concurrent Lines, Medians, and Altitudes Lesson 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. 5-3

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

Concurrent Lines, Medians, and Altitudes Lesson 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. 5-3

Concurrent Lines, Medians, and Altitudes Lesson 5-3 Additional Examples Finding the Circumcenter Find the center of the circle that circumscribes XYZ. 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 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 You need to determine the equation of two  bisectors, then determine the point of intersection. 5-3

Concurrent Lines, Medians, and Altitudes Lesson 5-3 Additional Examples (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. Quick Check 5-3

Concurrent Lines, Medians, and Altitudes Lesson 5-3 Additional Examples Real-World Connection City planners want to locate a fountain equidistant from three straight roads that enclose a park. Explain how they can find the location. The roads form a triangle around the park. 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. Quick Check 5-3

Circumcenter The point of concurrency of the perpendicular bisectors of the sides of a triangle.

Circumcenter The circumcenter is equidistant from each vertex of the triangle.

Incenter The point of concurrency of the three angles bisectors of the triangle.

Incenter The incenter is equidistant from the sides of a triangle.

Incenter The incenter is equidistant from the sides of a triangle.