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Chapter 5: Triangle Segments (MA.G.4.2 and MA.G.4.5) The 5 Segments we will discuss are: Midsegment (5.1) Perpendicular Bisector (5.2) Angle Bisector (5.3) Median (5.4) Altitude (5.4) After these segments are discussed we will then move on to relationships between the sides and angles of a triangle (5.6 and 5.7)

5.1: Midsegments of Triangles Chapter 5 is all about 5 special “SEGMENTS” that can be drawn inside a triangle Review Quickly: What’s a segment? What does a segment have to have in order to be a segment? These segments have special properties and allow us to find special points within the triangle that may be useful for various purposes. For example: one point can help you find the balancing point of the triangle, one point can help you find where to meet if 3 people are coming from different places

5.1: Midsegments of Triangles (MA.G.4.2 and MA.G.4.5) A Midsegment is a segment that connect the midpoints of two sides of a triangle Think: how many midsegments should I be able to draw in a triangle? If I draw all of them what have a created? Draw a Triangle ABC , Find the midpoints of all 3 sides and label them D, E, and F. Draw all 3 Midsegments and Identify the following: Which segments on the perimeter of the triangle are equal? What sides of the triangle are parallel to which midsegments? Special Properties of Midsegments: Triangle Midsegment Theorem: A midsegment will be half the length of the triangle side it is parallel too

5.2: Perpendicular and Angle Bisectors (MA.G.4.5) A Perpendicular Bisector has two important characteristics: It bisects a side (cuts it in half or goes through the midpoint) It makes a 90 degree angle with the side Think: how are midsegments similar to perpendicular bisectors and how are the two different? Draw a Triangle ABC , Find the midpoints of all 3 sides and label them D, E, and F. Draw all 3 Perpendicular Bisector and Identify the following: Which segments on the perimeter of the triangle are equal? Special Properties of Perpendicular Bisectors: Perpendicular Bisector Theorem: If a point is on the “PB” then it is equidistant from the endpoints of the segment. And Conversely, if a point is equidistant from the endpoints it must be on the perpendicular bisector.

Work in Groups to Solve the Following: Page 288 #9-25 Page 296 #16-23 CLASSWORK QUESTIONS Work in Groups to Solve the Following: Page 288 #9-25 Page 296 #16-23 EVEN ones will be graded next class for a HL Grade

5.3: Bisectors in Triangles http://www.khanacademy.org/math/geometry/triangles/v/circumcenter-of-a-triangle (8:00min) The Perpendicular Bisectors of the Triangle (We learned about them in 5.2) all meet at one point. Any time that lines meet, they intersect at a “POINT OF CONCURRENCY” The “point of concurrency” for perpendicular bisectors is called THE CIRCUMCENTER The CIRCUMCENTER has the special property that it is the same distance from each of the end points. The CIRCUMCENTER will be inside the triangle if the triangle is acute, on the hypotenuse if the triangle is right, and outside the triangle if the triangle is obtuse.

5.3: Bisectors in Triangles http://www.khanacademy.org/math/geometry/triangles/v/circumcenter-of-a-triangle The Angle Bisectors of a Triangle meet at a point of concurrency called the “INCENTER”. The Incenter has the special property that it is equidistant from each side of the triangle. It is also the center of a circle that has been inscribed in a triangle.

Work in Groups to Solve the Following: Page 305 # 7, 9, 15-18, 26, 28 CLASSWORK QUESTIONS Work in Groups to Solve the Following: Page 305 # 7, 9, 15-18, 26, 28

5.4: Medians and Altitudes The median of a triangle runs from a vertex to the midpoint of the opposite side. The 3 medians will meet at a point of concurrency called the CENTROID. The centroid has the special property that is 2/3 of the way from the vertex the opposite side. It is also called the balancing point. The altitude of a triangle runs from a vertex to a 90 degree angle on the opposite side. This is also known as the height of a triangle. The altitudes will meet at a point called the ORTHOCENTER

5.6: Triangle Comparison and Inequality Theorems The longest side of a triangle is always opposite the largest angle. Two sides of a triangle must always add to be bigger than the 3rd in order for the triangle to exist. Page 312 #8-13 Page 329 # 9, 13, 17, 19, 21, 25,

Work in Groups to Solve the Following: Page 312 # 8-13, 17, 19, 31 CLASSWORK QUESTIONS Work in Groups to Solve the Following: Page 312 # 8-13, 17, 19, 31