Transcript Aim:
Aim: What is the median, altitude and angle bisector of a triangle? Do Now: 1) What is the median of the numbers –4, -1, 0, 5, 6, 8, 10 ? Ans: 5 2) What is the altitude in miles of the plane in the diagram? Ans: 3 miles 3) If APB BPC , what can 4 miles P be said about PB ? Ans : PB is an angle bisector. A Geometry Lesson: Median, Altitude, Angle Bisector B C1 Def:A median of a triangle is a line segment that joins Def: Median any vertex of the triangle to the midpoint of the opposite side. Every triangle has three medians. Examples: Median • • • In each case below , make a congruence statement about 2 line segments. 2) VR is a 1) FL is a median in VQS median in EFG F E G L H P Q R EL LG V QR RS S T 2 Def:An altitude of a triangle is a line segment drawn from Def: Altitude any vertex of the triangle, perpendicular to and ending on the opposite side. Every triangle has three altitudes. Examples: Altitude In each case below, state the name of an altitude and the triangle to which it belongs. 1) A 2) B AL , ABD K L C D 3) BG , BNP P G N A C B Geometry Lesson: Median, Altitude, Angle Bisector B AC or BC , ABC Def:An Angle Def: angle bisector of a triangle is a line segment that Bisector bisects any angle of the triangle, and terminates on the side opposite the angle. Every triangle has three angle bisectors. Examples: Angle bisector Ex: PX is an angle bisector of APQ. Make a congruence statement about two angles. P APX XPQ L AXQ D Geometry Lesson: Median, Altitude, Angle Bisector Ex: Special line segments of triangles In each case, state whether TX is a median, altitude or angle bisector. T Q 2) Q 1) R T X X P TX is an altitutde in RTQ R P TX is a median in QTP T 3) TX is a median, altitude and angle bisector of RTP R X P Geometry Lesson: Median, Altitude, Angle Bisector Proofs w/Median, Altitude, Angle Bisector What conclusions can we make based on medians, altitudes and angle bisectors of triangles? 1) Given BD is an altitude of ABC : Conclusion: BD AC B C D A 2) Given BD is a median of ABC : Conclusion: D is midpoint of AC 3) Given BD is an angle bisector of ABC : Conclusion: ABD DBC Geometry Lesson: Median, Altitude, Angle Bisector 6 Proofs w/Median, Altitude, Angle Bisector T 1) Given: TN TL , N L TX is a median of NTL Prove: NTX LTX N 2) Given: TX is an altitude of NTL L X TX is an angle bisector of NTL Prove: NTX LTX 3) Given: TX is an altitude of NTL TX is a median of NTL Prove: NTX LTX Geometry Lesson: Median, Altitude, Angle Bisector 7 Proofs w/Median, Altitude, Angle Bisector T 4) Given: TX is a median of NTL TR TS , RN SL Prove: NTX LTX R• N Geometry Lesson: Median, Altitude, Angle Bisector •S X L 8