We think math is the greatest thing ever! What are they talking about? If it’s math, we already knew it was awesome!

What does it mean if lines are perpendicular? What is the symbol for perpendicular? What is a segment bisector? They intersect to form right angles.  A segment bisector intersects a line at its midpoint.

Use the vocabulary that you know to define perpendicular bisector. A perpendicular bisector intersects a line segment at its midpoint and is perpendicular to the line segment. bisector perpendicular midpoint Perpendicular – form right angles

Which diagram shows the perpendicular bisector of the side of a triangle?

Line a is a perpendicular bisector of ΔXYZ. X Y Z a 1 List three things that you know about the diagram. 2 T T is the midpoint of XZ. XT  ZT  1 and  2 are right angles a  XZ

Each side of a triangle has a perpendicular bisector. How many perpendicular bisectors does a triangle have? Let’s do this together Homework#6 Graph ΔABC with A (5, 5), B (15, 5) and C (11, 17). Find the perpendicular bisector of each side. Start with side AB.

AB C Midpoint of BC x-coordinate = y-coordinate = Midpoint is (13, 11) The slope of BC is Slope = -12/4 = -3/1 So, the slope of the  line is 1/3

A C B What do you notice about the three perpendicular bisectors of the sides of this triangle? Circumcenter

Graph ΔABC with A (5, 5), B (15, 5), and C (11, 17) Find the perpendicular bisectors of each side of the triangle. Use the chart below to organize your work. midpoint Slope of side Slope of line Perpendicular to side ABBCAC Graph the midpoint then count the slope of the perpendicular line. What are the coordinates of the circumcenter? (10,5)(13,11)(8,11) 0-3/1 2/1 undefined-1/3-1/2 (10,10)

Okay, let’s try another one.

Graph ΔLMN with L(-11, 4), M (-11, 14), and N (-1, 4) Find the perpendicular bisectors of each side of the triangle. Use the chart below to organize your work. midpoint Slope of side Slope of line Perpendicular to side LNLMMN Graph the midpoint then count the slope of the perpendicular line. What are the coordinates of the circumcenter?

Graph ΔXYZ with X (-11, -4), Y (-1, -14), and Z (11, -14) Find the perpendicular bisectors of each side of the triangle. Use the chart below to organize your work. midpoint Slope of side Slope of line Perpendicular to side YZXZXY Graph the midpoint then count the slope of the perpendicular line. What are the coordinates of the circumcenter?

What do you notice about the circumcenters of the triangles? How are the three triangles you graphed different from each other? Pat found the circumcenter of ΔEFG on the outside of the triangle. Sam classified ΔEFG as a right triangle. Pat and Sam’s teacher, Mrs. Geometry, asked if they were looking at the same triangle. What problem did Mrs. Geometry see? What can you conclude about the location of the circumcenter of a triangle ? Acute triangle - inside the triangle Obtuse triangle - outside the triangle Right triangle - on the hypotenuse of the triangle

y is the perpendicular bisector of a side of this triangle. Find the value for each variable. y 4x2x 3x (n² + 9)° 3y + 84y - 1

Find the value for each variable. Is line w a perpendicular bisector? w (8y)° n + 23n - 10 (9y - 7)° 3n - 2

Perpendicular bisectors / circumcenter A perpendicular bisector passes through the midpoint of a segment and is perpendicular to the segment The intersection of the three perpendicular bisectors of the sides of a triangle is the circumcenter of the triangle. circumcenter White Note Card Acute triangle – inside the triangle Right triangle – on hypotenuse Obtuse triangle – outside the triangle