Playground

1. Food for Though: Where should the park officials put a water fountain so that it is equidistant from the 3 locations? How should they go about this? Playground Tennis Court Volleyball Court

2. 5.3 part A Concurrent Lines, medians and altitudes LEQ: What are the properties of concurrent lines and how can we use them in problem solving?

3. When 3 or more lines intersect at one point, they are concurrent. • The point in which these lines intersect is called the point of concurrency. Point of concurrency

4. In this section: We will learn about 4 different types of points of concurrency: • Orthocenter • Incenter • Centroid • circumcenter Each of these is the intersection of different types of lines.

5. Draw a circle and construct 3 points (Q, S, and R) on the circle. • Connect the points to make a triangle. • As best you can, construct the perpendicular bisectors of each segment (they should intersect). • Label the point of intersection “C”. • This point is called the “circumcenter” of the triangle. • The circle is “circumscribed about” the triangle since Q,R, and S are equidistant from C. Q S C R

6. The point of concurrency of the perpendicular bisectors of a triangle is called the circumcenter of the triangle. • The circle connecting the vertices is “circumscribed about” the triangle. • The circle is “outside” the triangle and each of the vertices is “on” the circle.

7. Circumcenter: Perpendicular bisectors Circle is “circumscribed about”

8. U This time: X Y I • Draw triangle UTV. • The best you can, construct the angle bisectors of all three vertices. (these should intersect) • Label the point of intersection “I” • This point is called the “incenter” of the triangle • Drop a perpendicular line from I to each of the 3 sides. Label the points X,Y,Z as shown. • Draw a circle connecting these points. • The circle is “inscribe in” the triangle. V Z T

9. The point of concurrency of the angle bisectors of a triangle is called the “incenter.” • This time, the circle was “inscribed in” the triangle. • The circle is “inside” the triangle.

10. Incenter: Angle Bisectors Circle is “inscribed in”

11. Warmup: Using the picture below, identify the name of the point of concurrency. • Then grab a laptop and partner and log in.

12. Theorem 5-6 • The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices. • Theorem 5-7 • The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides. A YA=YC=YB Y B C H F P HP=PF=PC C

13. Circumcenter: Perpendicular bisectors Circle is “circumscribed about” Equidistant to vertices

14. Incenter: Angle Bisectors Circle is “inscribed in” Equidistant to sides of the triangle (must meet at right angles)

15. Where should the park officials put a water fountain so that it is equidistant from the 3 locations? How should they go about this? Playground Tennis Court Volleyball Court

16. Finding the circumcenter of the 3 points P(-2,3),O(-2,0) and S(2,0) • (0,1.5) Step 1: Graph the points and draw the triangle. (-2,3) P y Step 2: Draw the perpendicular bisectors of 2 sides…why? Step 3: Write the equations of the bisectors. S (2,0) x O (-2,0) Step 4: Find the point of intersection of the bisectors

17. Find the center of the circle that you can circumscribe about triangle ABC: • A(0,0) B(3,0) and C(3,2)

18. Homework • Pg. 275-276: 1-9,19,21,24 • No hwk passes

19. Lab: • Work with a partner to complete all questions (marked with a *) • Be specific with answers