Geometry IB Date: 2/18/2013 ID Check Objective: Students will identify and use medians and altitudes in triangles. Bell Ringer: Complete Constructions
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1 Geometry IB Date: 2/18/2013 ID Check Objective: Students will identify and use medians and altitudes in triangles. Bell Ringer: Complete Constructions HW Requests: pg 327 #17-25 odds 45, 46, pg 338 #11, 12 Exit Ticket: pg 338 #1-12 HW: Handout 5.1/5.2 Workbook pages from previous text Announcements:
2 Perpendicular Bisector Line, segment or ray that passes through the midpoint of the side and is perpendicular to that side. Circumcenter intersection of the 3 bisectors. The circumcenter is equidistant from the vertices. If O is the circumcenter OA 1 = OA 2 = OA 3.
3 Concurrent Lines: three or more lines intersect at a common point. Point of concurrency: point where concurrent lines intersect.
4 Median Median - segment whose endpoints are the vertex of an angle and the midpoint of the opposite side of the vertex. Centroid intersection of the 3 medians.. If O is the centroid, AO = ⅔ AP, BO = ⅔BQ, MO= ⅔MC Long = ⅔ total; Short = 1 3 total Long + Short = Total Median is associated with Midpoints.
5 Altitude Altitude segment from a vertex to the line containing the opposite side and perpendicular to the line containing that side. Orthocenter- intersection of 3 altitudes. Altitude is associated with right angles.
6 Summary Bisectors Circumcenter intersection of the 3 bisectors. The circumcenter is equidistant from the vertices. If O is the circumcenter OA = OB = OC. Medians Centroid intersection of the 3 medians. If O is the centroid, AO = ⅔ AP, BO = ⅔BQ. MO= ⅔MC Long = ⅔ total; Short = 1 3 total Long + Short = Total Altitudes Orthocenter- intersection of 3 altitudes.
7 Concurrent Lines: three or more lines intersect at a common point. Point of concurrency: point where concurrent lines intersect.
8 Name: Date: Per: Constructions pg 321 Materials: This sheet of paper Compass Protractor Straightedge Colored Pencils Exit Ticket 1:Pg 330 #46
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11 Median Median - segment whose endpoints are the vertex of an angle and the midpoint of the opposite side of the vertex. Centroid intersection of the 3 medians.. If O is the centroid, AO = ⅔ AP, BO = ⅔BQ, MO= ⅔MC Median is associated with Midpoints.
12 Geometry HR Date: 2/14/2013 ID Check 2nd Objective: SWBAT identify and use perpendicular and angle bisectors in triangles. Bell Ringer: Quiz Section minutes HW Requests: Quadratics WS 2 nd, Pg 327 #9-14, 21-26, 41, 42 If at first you don t succeed, try and HW: pg 327 #17-20, 27-30, try again. Announcements: Happy Valentines Day!
13 Geometry IB Date: 2/14/2013 ID Check 2nd Objective: SWBAT identify and use perpendicular and angle bisectors in triangles. Bell Ringer: Quiz Section minutes HW Requests: Pg 327 #9-14, 41, 42 HW: pg 327 #17-30 odds, 45, 46 Work on Personal Project! Announcements: Happy Valentines Day! If at first you don t succeed, try and try again.
14 Theorem 5.1 Perpendicular Bisector Theorem If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. A C P B perpendicular bisector If CP is the perpendicular bisector of AB, then CA = CB.
15 Theorem 5.2: Converse of the Perpendicular Bisector Theorem If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. A C P B If DA = DB, then D lies on the perpendicular bisector of AB. D is on CP D
16 Ex. 1 Using Perpendicular Bisectors In the diagram MN is the perpendicular bisector of ST. a. What segment lengths in the diagram are equal? b. Explain why Q is on M N MN. 12 T 12 Q S
17 Ex. 1 Using Perpendicular Bisectors a. What segment lengths in the diagram are equal? Solution: MN bisects ST, so NS = NT. Because M is on the perpendicular bisector of ST, MS = MT. (By Theorem 5.1). The diagram M N shows that QS = QT = T 12 Q b.explain why Q is on MN. Solution: QS = QT, so Q is equidistant from S and T. By Theorem 5.2, Q is on the perpendicular bisector of ST, which is MN. S
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19 Using Properties of Angle Bisectors Remember? The distance from a point to a line is defined as the length of the perpendicular segment from the point to the line. For instance, in the diagram shown, the distance between the point Q and the line m is QP. Q P
20 Theorem 5.4 Angle Bisector Theorem If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle. If m BAD = m CAD, then DB = DC A B D C
21 Theorem 5.5 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. If DB = DC, then m BAD = m CAD. A C B D Can you prove that BD = CD, using what you know about triangles?
22 Ex. 3: Using Angle Bisectors Roof Trusses: Some roofs are built with wooden trusses that are assembled in a factory and shipped to the building site. In the diagram of the roof trusses shown, you are given that AB bisects CAD and that ACB and ADB are right angles. What can you say about BC and BD? O C A D B M L G H K N P
23 SOLUTION: Because BC and BD meet AC and AD at right angles, they are perpendicular segments to the sides of CAD. This implies that their lengths represent distances from the point B to AC and AD. Because point B is on the bisector of CAD, it is equidistant from the sides of the angle. So, BC = BD, and you can conclude that BC BD. O C A D B M L G H K N P
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25 Perpendicular Bisector A segment, ray, line, or plane that is perpendicular to a segment at its midpoint is called a perpendicular bisector. Perpendicular Bisectors in a triangle
26 1. Rotation 2. Reflection 10 minutes
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28 What is the best way to track the constellations? How does GPS work?
29 Placing Triangles on coordinate plane Key Concept pg 301 Step 1: Use the origin as a vertex or center of the triangle Step 2: Place at least one side of a triangle on an axis. Step 3: Keep the triangle within the first quadrant, if possible. Step 4: Use coordinates that make computations as simple as possible.
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32 Geometry HR Date: 2/8/2013 ID Check Objective: Identify reflections, translations, an rotations and verify congruence after a congruence transformation. Bell Ringer: See overhead HW Requests: pg 297 #7-23 odds Parking Lot: Perfect Square Trinomials, OEA #33 In class: Graph pg 298 #17-20, HW: Quadratic WS (Half Sheet) Announcements: Quiz Section Monday If at first you don t succeed, try and try again.
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34 Geometry IB-HR Date: 2/7/2013 ID Check Objective: Identify reflections, translations, an rotations and verify congruence after a congruence transformation. Bell Ringer: Go over Red WB Sect. 4.6 HW Requests: pg 287 #9-21 odds, 29-32, 38, OEA #33 Parking Lot: Perfect Square Trinomials In class: Take Cornell Notes Pg 297 #1-6, pg 299 #24-26, 32 HW: pg 297 #7-23 odds Exit Ticket: pg 299 #24-26, 32 Announcements: Quiz Section Monday If at first you don t succeed, try and try again.
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49 Geometry IB -HR Date: 2/4/2013 ID Check Objective: Use properties of isosceles and equilateral triangles. Bell Ringer: Put OEA in Bin - Go over OEA #46. HW Requests: Pg 291 #52-55 In class: Take Cornell Notes HW: pg 287 #9-21 odds, 29-32, 38, OEA #33; Read Sect. 4.7 If at first you don t Exit Ticket: succeed, try and try again. Announcements: Quiz Section Monday Selected Problems pg 287 #1-7
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51 Properties of Isosceles Triangles Vertex Angle The angle formed by the congruent sides. Base Angle Two angles formed by the base and one of the congruent sides.
52 Thm Isosceles Triangle Thm. If two sides of a triangle are congruent, then the angles opposite those sides are congruent. Ex: Proof 1
53 Thm Converse of Isosceles Triangle Theorem If two angles of a triangle are congruent, then the sides opposite those angles are congruent. Ex: Proof
54 Equilateral Triangles Corrollary 4.3 A is equilateral if and only if it is equiangular. Corrollary 4.4 Each angle of an equilateral measures 60 degrees.
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Geometry HR Date: 2/13/2013 ID Check 2nd Objective: SWBAT identify and use perpendicular and angle bisectors in triangles.
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