University:
Felician University
Course:
MATH-231 | Geometry I
Academic year:

2020
Views:

210

Pages:

11
Author:

SpaceCadet101

• Segments in Triangles Obiective: *Know what а Mid-Segment is and it's properties: 1) а Mid-segments is Parallel to it's partner side 2) а Mid-Segment is ½ as long as it's partner side *Know what а Perdendicular Bisector is and it's Properties: *Know what а Angle Bisector is and it's Properties: *Know what а Median is and it's Properties: *Know what а Altitude is and it's Properties: *Know the names of the centers created Ьу these segments Mid-segments (one quick new triangle property) • Midsegments are line segments created Ьу connecting the midpoints of two sides of а triangle. Х is the midpoint of АВ У is the midpoint of ВС ХУ is а midsegment • Every triangle has three midsegments. Е Е Е Triangle Midsegшent Theorern А шidsegшcnt of а triangle is paralle\ to t\1e tl1ird side and is l1alf as loпg. в Examplc 1 Fiпd the lengths of В(· and Y.l and the n,easure of LAX.7.. А 'ЬС. ::: � �:r, e,G := ZL4-� Q,C � �-Ч> '1� < = 2 �'!:, 1� -= 2lt\.s) \ Y:z::= 6-1'? l z с Mark what you know! (angles and segments) Example2 Jf АВ = 2\' + 7 and YZ = Зх- 6.5, ,\·hat is tl1e \ength of АВ? А Ask yourself 'lf YZ is 5, how long is АВ?' How did you decide the length of АВ? Use that pattern. '2. ·1-. �8 �=- ( Ъk �.с:; -= '2 -\- -=1- � - 1� � 2:�-\-81-- ч к-:::: 2,,0 Because Mid-Segments are // to their partner sides we сап use angle pairs to solve proЫems. А в с Mark what you know ! (angles and segments) I\Лorli�in• Г"\111 Л li.-.1e VI line serтme nr JIII nf er e c-irle in+\.er n+ +ha+ +c- +h Б 1 11\. \.11 L II 1s \..\..З f..11 ..:>IU VI IYIC'UIOII. а triangle from one angle to the midpoint of the opposite side. lmportant idea #1: Median goes from angle to midpoint. When we draw three of the above segments we get а 'Point of Concurrency': The point at which three or тоге lines intersect. Let's identify the point of concurrency for medians and learn some of their properties ! ! ! The intersection of the medians is called the CENTROID. Centroid Theorem 2 з 111е ce11troid ofа t·riar}gle is - tl}e dista11ce froш each vertex to tl1e midpoint ofthe opposite side. х + 2х = whole с The ce11troid oftltis tria11g!e is at point Х. 2 2 2 ЛХ=-ЛU • ВХ=-ВV· СХ=-СТ з их = 3 , • з ц; ""' • ::: з iвv ; •¼-=�t l:.T practice PointG is tbe ce11troid of ЛАВС,АD = 8,AG = 10,ВЕ = 10,АС = 16 and CD = 18. Find tbe lengtb of each segment. в 17). DB=_B __ 1s). ЕА=_r_ъ __ А с F Point G is the ce11troid of Л АВС. Use the given infonnation to find thevalue of the variaЫe. в 19) f'G= 2х + 8 andGA= бх-4 ;z(-z�+8) = lPx-+ 4x-t Но.::: (o.X-t.f Ц) -= '2'f.. А о 10 Altitude: А line or line segment that intersects the side of а triangle at а 90 degree angle. {lt goes from an angle and is perpendicular to the opposite side.) *The point of concurrency for 3 altitudes is called an Orthocenter. А в с lmportant idea: Altitude goes from angle and is perpendicular to opposite side (does not have to go to the midpoint!) 11 Perpendicular Bisector: А line that passes through the midpoint of the side of а triangle at а 90 degree angle. {lt goes from the middle of а side and is perpendicular to that side.) ..L Picture: Both sides ai·e congruent- make sure you see this or it is NOT а perpendicular • The point of concurrency for three perpendicular blsectors is call the circumcenter. Property One: The distance from the CIRCUMCENTER го each vertice of the triangle is the same! {see next slide). fn.ьм --Т-о C,,\�W\�\JТ�L \J�Т'\(.,e,S ДLL �\)\ О \SТ--NJ'\ 13 Practice examples 1n tbe diagram, the perpeodiculat· Ьisectors of Л.4ВС meet at point G-tbe circumce11t.er. aod are sbown dasbed. Find the indicated measure. i2) AG= 9.5 в IЗ)BD=-�-- А 14). IFBG=(2x-15),findx. P.:,G = АС:5 ==/v 24 F 2Ч -� (_ & 14 City Planning �lю\,· \\'IH·rt .· to\,·o ol'lki,,I.:.. -.l111t1l�I рl.н t· а rt·cycliн� l)aпc·I �о tl1a1 it i, c·чнis мь�, А-Т цQ..�uм�� wr\t� Ь-\ � � t \SrfЧJf Lif u rd chltir 'f D U- V�\\ Ce.-S 15 ,...., Angle Bisector: А line or line segment that divides the angle into two equal parts. (lt goes from the angle, to some point on the opposite side.) R The point ot· concurrency for three angle Ьisectors is call the incenter. Property One: The distance from the INCENTER to each side of the triangle is the same! (see next slide). 16 lncenter Theorem The incenter of а triangle i� equidistance fr9m the sides of а triangle. The incenter of this trianglt is at Х. XT=XU=XV 17 practice P_____w......,...____o Point Пs the i1icenter of ЛPQR. ! 15 15) ST= JS '2..Ч о 16) IfmLPRT=24 °, thenmLQRT= __•__ 18 Review - What is segment АВ? 1) 2) 6) в Coordinate Geometry в MбDLAtJ 1:i,нl tlн· t ooпli11.1tt·, о! tlн· ( �п tш1u•11tt·r tlн·111.111�l('\\itl1\·t·111�t•,,\( ·1,:'»).Н( 2,.i).a,нl(( 2, :!). 19 01· Coordinate Geometry t·11нl tlн· t 1111нl111.1te, 11! tlн· пr� шнc(·ritt·r 111 tlн· tr 1a11�lt· у J ,:; ✓ * lt> - Ч � d::: ✓ llP+tft> -:::� 2. l 11-0 � 2- � и� Q:r �1.дRб L6 С\ f2L,t;> елд с_ G �� -:::=-) {_ь 1D \,\"!Q t1fE,t-) �� 'l- "Z.. �.f .(32 3� ..\- 3 µ - tp � G:k\-r 1""lli 21 � cf 6Lб \\'l1at ,tr • tlн· ( (1or(1irцtt·'- ol tlн· 1r< ш1н ·11ter of 1lн· rr1.1r1�lt· \\ 1111, cr 1н t· \("l.1). Щ 1011) .. ннl l ( \О,'3)( '2 1 5 - .з -- 6 Уе) 'L c1u.vм&u1oo. с(rJ ) 5 22 � [е_,