Chapter 5

Vocab Review  Intersect  Midpoint  Angle Bisector  Perpendicular Bisector  Construction of a Perpendicular through a point on a line Construction of a Perpendicular through a point on a line

Perpendicular Bisector Theorem If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment

Converse of the Perpendicular Bisector Theorem If a point is equidistant from the endpoints of the segment, then it is on the perpendicular bisector of a segment

Using Perpendicular Bisectors  What segment lengths are equal?  Why is L on OQ? Given: OQ is the  bisector of MP

Angle Bisector Theorem If a point is on the bisector of an angle, then it is equidistant from the two side of the angle

Converse of the Angle Bisector Theorem If a point is in the interior of an angle AND is equidistant from the sides of the angle, then it lies on the bisector of the angle

Angle Bisector Theorem Let’s Prove it… Given: D is on the bisector of  BAC BD  AB CD  AC Prove: DB  DC STATEMENTSREASONS

“Guided Practice” Page 267 #1-7 CHECK POINT

Perpendicular Bisector of a Triangle Line (or ray/ segments) that is perpendicular to a side of the triangle at the midpoint of the side. Circumcenter (not necessarily inside the triangle)

Perpendicular Bisector of a Triangle

Concurrency  Point of Concurrency  The point of intersection of the lines  Concurrent Lines  When 3 or more lines (rays/segments) intersect at the same point.

Concurrency of Perpendicular Bisectors of a Triangle The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle AG = BG = CG Circumcenter

Exit Ticket

Angle Bisector of a Triangle Bisector of an angle of the triangle. (one for each angle) Incenter (always inside the triangle)

Angle Bisector of a Triangle

Concurrency of Angle Bisectors of a Triangle The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle EG = DG = FG Incenter

Pythagorean Theorem a 2 + b 2 = c 2

Pythagorean Theorem a 2 + b 2 = c 2 EXAMPLE: a= ?, b= 15, c= 17 a 2 + b 2 = c 2 a = 17 2 a = 289 a 2 = 64 a = 8

Classwork Page #10-23, CHECK POINT

Median of a Triangle Segment whose end points are a vertex of the triangle and the midpoint of the opposite side. Centroid (always inside the triangle)

Median of a Triangle

Concurrency of Medians of a Triangle The medians of a triangle intersect at a point that is ²/₃ of the distance from each vertex to the midpoint of the opposite side. AG = ²/₃AD, BG = ²/₃BE, CG = ²/₃CF Centoid

Page 280 Example #1 EXAMPLE

Altitude of a Triangle Perpendicular segment from a vertex to the opposite side. Orthocenter (inside, on, or outside the triangle)

Page 281 Example #3 EXAMPLE

Concurrency of Altitudes of a Triangle Lines containing the altitudes of a triangle are concurrent Intersect at some point (H) called the orthocenter (Inside, On, or Outside) orthocenter

Activity Video

Classwork: pg 282 #13-16 Homework: pg 282 #1-11, 17-20, & 34 CHECK POINT

Warm-up Find the coordinates of the midpoint: 1. (2, 0) and (0, 2) 2. (5, 8) and (-3, -4) Find the slope of the line through the given points: 1. (3, 11) and (3, 4) 2. (-2, -6) and (4, -9) 3. (4, 3) and (8, 3)

Midsegment of a Triangle *Segment that connects the midpoints of two sides of a triangle *Half as long as the side of the triangle it is parallel to

** REVIEW ** MIDPOINT FORMULA Midpoint =

Midsegment of a Triangle  Find a Midsegment of the triangle…. Let D be the midpoint of AB Let E be the midpoint of BC

Midsegment Theorem The segment connecting the midpoint of two sides of a triangle is parallel to the third side and is half as long DEBC and DE=¹⁄₂BC

Prove it… How do you prove DEBC? How do you prove DE=¹⁄₂BC? Use the slope formula to find each slope (parallel lines have the same slope) Use the distance formula to find the distance of each segment

Exit Ticket Pg 290 #3-9

Classwork: pg 290 #10-18, CHECK POINT

Warm-up In ∆DEF, G, H, and J are the midpoints of the sides… J H G F ED 1. Which segment is parallel to EF? 2. If GJ=3, what is HE? 3. If DF=5, what is HJ? GH 3 2.5

Pgs #1-15 Pgs 313 #1-11 Test Review