Playground Volleyball Court Tennis Court

LEQ: What are the properties of concurrent lines and how can we use them in problem solving?

 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

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.

 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. QS R C

 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.

Circumcenter: Perpendicular bisectors Circle is “circumscribed about”

 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. U T V I X Y Z

 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.

Incenter: Angle Bisectors Circle is “inscribed in”

 Then grab a laptop and partner and log in.

 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 B C Y YA=YC=YB HF C PHP=PF=PC

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

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

Playground Volleyball Court Tennis Court

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

 A(0,0) B(3,0) and C(3,2)

 Pg : 1-9,19,21,24  No hwk passes

 Work with a partner to complete all questions (marked with a *)  Be specific with answers