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1、CONCURRENT LINES, MEDIANS & ALTITUDES茅辙佑瞧酸烂佛闸赣挂刘面境赁蝇馈菱峙沫貌家煮扬捅瞥颁收柏瘸坚橱恼Concurrent Lines, Medians & Altitudes - CunninghamWAHS - 行线，中线&；海拔cunninghamwahsConcurrent Lines, Medians & Altitudes - CunninghamWAHS - 行线，中线&；海拔cunninghamwahsVocabularyConcurrent Lines when three or more lines inter
2、sect in one point.Point of concurrency the point at which 3 or more lines intersect.磺障诱莽饱铬仲匝梁翅瘁津嗣峡秧称捌壬痈弧所售精伞魂鬼杨霖钵幸媳狗Concurrent Lines, Medians & Altitudes - CunninghamWAHS - 行线，中线&；海拔cunninghamwahsConcurrent Lines, Medians & Altitudes - CunninghamWAHS - 行线，中线&；海拔cunninghamwahsGeogebra D
3、emonstration of Perpendicular Bisectors枕诱姿瘸专装般恼烷饱苹瞥症绢灵寡淌擞龚邦钉焰押泼乒稀挡蔬贯赛措未Concurrent Lines, Medians & Altitudes - CunninghamWAHS - 行线，中线&；海拔cunninghamwahsConcurrent Lines, Medians & Altitudes - CunninghamWAHS - 行线，中线&；海拔cunninghamwahsVocabularyCircumcenter of the triangle the point of con
4、currency of the perpendicular bisectors.Circumcenter彝伦秉嫁未右溜踞慢罕椎垦息辉氓棠卉勇玄阶债锅辑盖槛默呢管锄小碎朵Concurrent Lines, Medians & Altitudes - CunninghamWAHS - 行线，中线&；海拔cunninghamwahsConcurrent Lines, Medians & Altitudes - CunninghamWAHS - 行线，中线&；海拔cunninghamwahsThe perpendicular bisectors of the sides o
5、f a triangle are concurrent at a point equidistant from the vertices.Theorem男红闸犹赦癸兹篙萧乖淖意拦厚防园忠理尤陆宝妙狡奢伟鸽袍辐烧泪危壕Concurrent Lines, Medians & Altitudes - CunninghamWAHS - 行线，中线&；海拔cunninghamwahsConcurrent Lines, Medians & Altitudes - CunninghamWAHS - 行线，中线&；海拔cunninghamwahs Since the vertice
6、s of the triangle are equidistant from the circumcenter, we can draw a circle around the triangle or circumscribe the triangle.The center of the circle is the circumcenter of the triangle.熄姨迸文砾奶长尽融矗椽闪轿犊伴牛习惫挪让鲤矩村奄钙钞苑丰列涤干据Concurrent Lines, Medians & Altitudes - CunninghamWAHS - 行线，中线&；海拔cunning
7、hamwahsConcurrent Lines, Medians & Altitudes - CunninghamWAHS - 行线，中线&；海拔cunninghamwahsGeogebra Demonstration of Angle Bisectors课锁癌律租珊亭官首篡饲丛至校波蝶缅份馋乾蛰挡洒智谎榆煎铸凯诌荷决Concurrent Lines, Medians & Altitudes - CunninghamWAHS - 行线，中线&；海拔cunninghamwahsConcurrent Lines, Medians & Altitudes - Cu
8、nninghamWAHS - 行线，中线&；海拔cunninghamwahsVocabularyIncenter of the triangle the point of concurrency of the angle bisectors.Incenter沁稀沿歧座衰瑚咋壳斡费淑洗瑚主蚁悍曲碟漆之删滨趟铬获孙台挨杠杰惜Concurrent Lines, Medians & Altitudes - CunninghamWAHS - 行线，中线&；海拔cunninghamwahsConcurrent Lines, Medians & Altitudes - Cunni
9、nghamWAHS - 行线，中线&；海拔cunninghamwahsTheoremThe bisectors of the angles of a triangle are concurrent at a point equidistant from the sides.党希键帐咬痞依腆渺苔绳惺乓诈网纠阶娥碘刀诛摆狠高剂狄枣蜡坷渺憋眺Concurrent Lines, Medians & Altitudes - CunninghamWAHS - 行线，中线&；海拔cunninghamwahsConcurrent Lines, Medians & Altitudes
10、 - CunninghamWAHS - 行线，中线&；海拔cunninghamwahs We can now inscribe a circle in the triangle since the incenter is equidistant from the sides.The center of the circle is the incenter of the triangle.谭袒厄侮氏踌壮扼蓟文项磊盯码咐腑胆樱狰盟敲胰授侗囤苗部吊歇疮利抢Concurrent Lines, Medians & Altitudes - CunninghamWAHS - 行线，中线&
11、；海拔cunninghamwahsConcurrent Lines, Medians & Altitudes - CunninghamWAHS - 行线，中线&；海拔cunninghamwahsGeogebra Demonstration of Altitudes of Triangles机述籍漳暇锨帧终壹荣阉察政革辛曹弦碘崭遥诗昼佩侦嫡猾烬蔑域挚斋粕Concurrent Lines, Medians & Altitudes - CunninghamWAHS - 行线，中线&；海拔cunninghamwahsConcurrent Lines, Medians &am
12、p; Altitudes - CunninghamWAHS - 行线，中线&；海拔cunninghamwahsVocabularyAltitude of a Triangle a perpendicular segment from a vertex to the line containing the side opposite to the vertex.Orthocenter函伤庸融临真每捍睬斯铲混奔效辉吃朴门赐蒸蔗随辩局乳寡两靠翌镭猎紧Concurrent Lines, Medians & Altitudes - CunninghamWAHS - 行线，中线&；海拔
13、cunninghamwahsConcurrent Lines, Medians & Altitudes - CunninghamWAHS - 行线，中线&；海拔cunninghamwahsFYIAn altitude can be the side of the triangle.呐幅洋簇秒济涸位崭卵蛔锤栋呕怖缕眷惯魄湾兽陛注簿箔刹愁区葫线绢女Concurrent Lines, Medians & Altitudes - CunninghamWAHS - 行线，中线&；海拔cunninghamwahsConcurrent Lines, Medians & A
14、ltitudes - CunninghamWAHS - 行线，中线&；海拔cunninghamwahsTheoremThe lines that contain the altitudes of a triangle are concurrent.谨八缉桃逆衍磐涵炒给撕家附扣颧撅掉姬摆包垃世助快宿飘蓉搂任埂泰运Concurrent Lines, Medians & Altitudes - CunninghamWAHS - 行线，中线&；海拔cunninghamwahsConcurrent Lines, Medians & Altitudes - Cunningham
15、WAHS - 行线，中线&；海拔cunninghamwahsGeogebra Demonstration of Medians of Triangles劲霓恫滴阜闹屈乒载整亩赣练乱尽壕傅蛀伶晌抹漳搀萝札超碟汁宵各挖闯Concurrent Lines, Medians & Altitudes - CunninghamWAHS - 行线，中线&；海拔cunninghamwahsConcurrent Lines, Medians & Altitudes - CunninghamWAHS - 行线，中线&；海拔cunninghamwahsVocabularyMedia
16、n of a Triangle a segment whose endpoints are a vertex and the midpoint of the opposite side.Centroid执九百胁仗寥火抡物扰辕陡猴锻匿西宙莉抹染鄂夺囊漾霓遏颁活爆幢亮识Concurrent Lines, Medians & Altitudes - CunninghamWAHS - 行线，中线&；海拔cunninghamwahsConcurrent Lines, Medians & Altitudes - CunninghamWAHS - 行线，中线&；海拔cunning
17、hamwahsFYIThe centroid is the center of gravity of the triangle. If you cut out the triangle and paper-fold to determine the centroid, you can balance the triangle on the tip of your pencil point. 艘琳临册富福姿湾状竹乞愿伦雅炙宇国座郴啮韧虽酶颜科兜快暑瘸瞅芜惕Concurrent Lines, Medians & Altitudes - CunninghamWAHS - 行线，中线&；
18、海拔cunninghamwahsConcurrent Lines, Medians & Altitudes - CunninghamWAHS - 行线，中线&；海拔cunninghamwahsTheoremThe medians of a triangle are concurrent at a point that is two thirds the distance from each vertex to the midpoint of the opposite side.榔希面射胎酣凡未秃爸晦寥话眶份爹析吉骂草吞睦谱扇留迪石苗彰屠捐脓Concurrent Lines, Me
19、dians & Altitudes - CunninghamWAHS - 行线，中线&；海拔cunninghamwahsConcurrent Lines, Medians & Altitudes - CunninghamWAHS - 行线，中线&；海拔cunninghamwahsABCDEFG越交肄析嘎许东毅柠两妮毁林订冲积帐栏郝脐档唤竭秋膜衅级乒板医倪盐Concurrent Lines, Medians & Altitudes - CunninghamWAHS - 行线，中线&；海拔cunninghamwahsConcurrent Lines, Me
20、dians & Altitudes - CunninghamWAHS - 行线，中线&；海拔cunninghamwahsCut out your triangles. Yellow Paper - MedianDraw an acute triangle.Cut it out.Determine the midpoint of each side by folding corner to corner. Then fold the triangle from the midpoint of each side to the vertex opposite it.Ask Mrs.
21、Slifko to see the model in my notebook if you need to .Label the point of concurrency the centroid. Try to balance your triangle on the tip of your pencil.Put any other theorems on the triangle.诧鱼粟掳肋都军鸿捉钎怎抒盈寡母禾快剐象掉禽宗补豫锚疾酬瓢侗延盔石Concurrent Lines, Medians & Altitudes - CunninghamWAHS - 行线，中线&；海拔c
22、unninghamwahsConcurrent Lines, Medians & Altitudes - CunninghamWAHS - 行线，中线&；海拔cunninghamwahsCut out your triangles. Green Paper- AltitudeDraw an acute triangle.Cut it out.Fold the triangle so that you create an altitude extending from all three vertices.Ask Mrs. Slifko to see the model in my
23、 notebook if you need to .Label the point of concurrency the orthocenter. Write any necessary information on the triangle.裤哉顽蜘夺禄寓邵共鼻炸口供劫糙妮嘴郝铰煎炊捌蓟寒糯怀痈附冀驾棵闭Concurrent Lines, Medians & Altitudes - CunninghamWAHS - 行线，中线&；海拔cunninghamwahsConcurrent Lines, Medians & Altitudes - CunninghamWAHS
24、- 行线，中线&；海拔cunninghamwahsWe can use coordinate geometry to write equations of the various special segments of the triangles.奉沪渭乎揖烩昌见汛拆骄捂生饱编借匪竿撞糠张甲盅沽卓谐怀柜舱领旱页Concurrent Lines, Medians & Altitudes - CunninghamWAHS - 行线，中线&；海拔cunninghamwahsConcurrent Lines, Medians & Altitudes - Cunningham
25、WAHS - 行线，中线&；海拔cunninghamwahsExample:A(2,9)B(-2,3)C(6,-1)How can we find the slope of altitude AD?DWe know that altitudes are at right angles to the opposite sidestherefore, if we can find the slope of BC, we can use opposite reciprocals to find the slope of AD.紫内晤踪酗赠妆汐提测噎慢嘎郎巫奏努巧败前颈克患音峡粗动金惨科涝曳Co
26、ncurrent Lines, Medians & Altitudes - CunninghamWAHS - 行线，中线&；海拔cunninghamwahsConcurrent Lines, Medians & Altitudes - CunninghamWAHS - 行线，中线&；海拔cunninghamwahsExample:A(2,9)B(-2,3)C(6,-1)How can we find the equation of altitude AD?DIf you know a point on the altitude , namely A(2,9),and
27、 the slope of AD, then you can use the point slope formula to find the equation of altitude AD.也荔橇川呆结微索醚墒闷拍湘就共晚辑谦育际福禄修偿淹遇陨翁桑仇周痒Concurrent Lines, Medians & Altitudes - CunninghamWAHS - 行线，中线&；海拔cunninghamwahsConcurrent Lines, Medians & Altitudes - CunninghamWAHS - 行线，中线&；海拔cunninghamwah
28、sExample:A(3,9)B(-2,3)C(6,-1)How can we find the slope of median AD?DSince we know that a median hits the midpoint, we know that D is the midpoint of BC. Once you find the midpoint, you can use the slope formula to calculate the slope of AD.笛聚雷毖复杆蘸片煽设火泽经麦铀仑吩到煌筛聊蒜妇哪曼忿专山窍够疮设Concurrent Lines, Medians &
29、amp; Altitudes - CunninghamWAHS - 行线，中线&；海拔cunninghamwahsConcurrent Lines, Medians & Altitudes - CunninghamWAHS - 行线，中线&；海拔cunninghamwahsExample:A(3,9)B(-2,3)C(6,-1)How can we find the equation of median AD?DSince you know a point on the median line and a slope, you can use point-slope for
30、mula to determine the equation of median AD.蛙号蹈驯伺荐档师醛丽侯泉燥剃沽导矽肿杯勾担仲武赌巡盛弯明北陆纽绽Concurrent Lines, Medians & Altitudes - CunninghamWAHS - 行线，中线&；海拔cunninghamwahsConcurrent Lines, Medians & Altitudes - CunninghamWAHS - 行线，中线&；海拔cunninghamwahsExample:A(6,9)B(-2,3)C(12,-1)How can we find the s
31、lope of the perpendicular bisector of BC?DSince we know that perpendicular bisectors form right angles, we can determine the slope of BC and then use opposite reciprocals to determine the slope of the perpendicular bisector of BC.耿洞憨巴页戏摇匠死证爽疲沥乞戍失驰剧迄存冻对展并押却傣尸卢畏暖绪Concurrent Lines, Medians & Altitudes - CunninghamWAHS - 行线，中线&；海拔cunninghamwahsConcurrent Lines, Medians & Altitudes - CunninghamWAHS - 行线，中线&；海拔cunninghamwahs
