MeasuringSegments

Unit1Lesson3

MeasuringSegments

Studentswillbeableto:

•Findthemeasureofsegmentsusingtherulerpostulate,thedefinitionofcongruentsegments,midpointsandthesegment

additionpostulate.• Findthelengthandmidpointofasegment.• Understandthecongruencyofsegments.

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MeasuringSegments

KeyVocabulary:LineSegmentDistance/length

CongruentsegmentsSegmentbisectorSegmentpartition

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MeasuringSegments

• Alinesegment isasetofpointsandhasaspecificlengthi.e.itdoesnotextendindefinitely.• Alinesegmentisnamedbytwopointsonthelinesegmentwithalinesegmentsymbolabovetheletters.• Thelinesegmentalwayshasacertainlengththatcanbemeasuredusingaruler.

MeasuringSegments

RulerPostulate• Everypointonalinecanbepairedwitharealnumber.Thismakesaone-to-onecorrespondencebetweenthepointsonthelineandtherealnumbers.• Therealnumberthatcorrespondstoapointiscalledthecoordinateofthepoint.

MeasuringSegments

SampleProblem1: Findthelengthofeachsegmentusingaruler.a.

𝑨𝑩

𝑨𝑩 =?

MeasuringSegments

SampleProblem1: Findthelengthofeachsegmentusingaruler.a.

𝑨𝑩

𝑨𝑩 =?

𝑨𝑩 = 𝐴𝑐𝑡𝑢𝑎𝑙𝑙𝑒𝑛𝑔𝑡ℎ𝑓𝑟𝑜𝑚𝑦𝑜𝑢𝑟𝑟𝑢𝑙𝑒𝑟

MeasuringSegments

SampleProblem1: Findthelengthofeachsegmentusingaruler.b.

𝑻

𝑻𝑷 =?𝑷

MeasuringSegments

SampleProblem1: Findthelengthofeachsegmentusingaruler.b.

𝑻

𝑻𝑷 =?

𝑻𝑷 = 𝐴𝑐𝑡𝑢𝑎𝑙𝑙𝑒𝑛𝑔𝑡ℎ𝑓𝑟𝑜𝑚𝑦𝑜𝑢𝑟𝑟𝑢𝑙𝑒𝑟

𝑷

MeasuringSegments

SampleProblem1: Findthelengthofeachsegmentusingaruler.c.

𝑹

𝑹𝑫 =?

𝑫

MeasuringSegments

SampleProblem1: Findthelengthofeachsegmentusingaruler.c.

𝑹

𝑹𝑫 =?

𝑹𝑫 = 𝐴𝑐𝑡𝑢𝑎𝑙𝑙𝑒𝑛𝑔𝑡ℎ𝑓𝑟𝑜𝑚𝑦𝑜𝑢𝑟𝑟𝑢𝑙𝑒𝑟

𝑫

MeasuringSegments

Findingthelengthoflinesegmentusingnumberline• Thedistancebetweenpointsonnumberlinerepresentthelengthoflinesegment.• Thedistancebetweenanytwopointsistheabsolutevalueofthedifferenceofthecoordinates.Ifthecoordinatesofpoints𝑨 and𝑩 are𝒂 and𝒃,thenthedistancebetween𝑨 and𝑩 is:𝑨𝑩 = 𝒂 − 𝒃 or𝑨𝑩 = 𝒃 − 𝒂• Twolinesegmentshavingsamelengthsaresaidtobecongruentlinesegments.

MeasuringSegments

SampleProblem2: Findthelengthofeachsegmentusingnumberline.Determinewhichsegmentsarecongruent.a. 𝑨𝑪 =? 𝑪𝑷 =? 𝑯𝑳 =? 𝑨𝑳 =?

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𝑨𝑪𝑯𝑷𝑳

MeasuringSegments

SampleProblem2: Findthelengthofeachsegmentusingnumberline.Determinewhichsegmentsarecongruent.a. 𝑨𝑪 =? 𝑪𝑷 =? 𝑯𝑳 =? 𝑨𝑳 =?

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𝑨𝑪𝑯𝑷𝑳

𝑨𝑪 = −𝟓 − −𝟐 = −𝟓 + 𝟐 = −𝟑 = 𝟑𝑪𝑷 = −𝟐 − 𝟒 = −𝟔 = 𝟔𝑯𝑳 = 𝟐 − 𝟓 = −𝟑 = 𝟑𝑨𝑳 = −𝟓 − 𝟓 = −𝟏𝟎 = 𝟏𝟎

𝑨𝑪 ≅ 𝑯𝑳𝑨𝑪 = 𝑯𝑳

MeasuringSegments

SampleProblem2: Findthelengthofeachsegmentusingnumberline.Determinewhichsegmentsarecongruent.b. 𝑱𝑼 =? 𝑹𝑻 =? 𝑮𝑹 =? 𝑭𝑻 =?

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𝑹𝑮𝑻𝑭𝑼𝑱

MeasuringSegments

SampleProblem2: Findthelengthofeachsegmentusingnumberline.Determinewhichsegmentsarecongruent.b. 𝑱𝑼 =? 𝑹𝑻 =? 𝑮𝑹 =? 𝑭𝑻 =?

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𝑹𝑮𝑻𝑭𝑼𝑱

𝑱𝑼 = 𝟓 − 𝟑 = 𝟐 = 𝟐𝑹𝑻 = −𝟔 − (−𝟏) = −𝟔 + 𝟏 = −𝟓 = 𝟓𝑮𝑹 = −𝟒 − (−𝟔) = −𝟒 + 𝟔 = 𝟐 = 𝟐𝑭𝑻 = 𝟐 − (−𝟏) = 𝟐 + 𝟏 = 𝟑 = 𝟑 𝑱𝑼 ≅ 𝑮𝑹

𝑱𝑼 = 𝑮𝑹

MeasuringSegments

SegmentAdditionPostulate:

Ifthreepoints𝑨,𝑩,and𝑪 arecollinearand𝑩 isbetween𝑨 and𝑪,thenthedistance𝑨𝑪 isthesumofdistance𝑨𝑩 and𝑩𝑪.

𝑨𝑩 + 𝑩𝑪 = 𝑨𝑪

MeasuringSegments

SampleProblem3: Findthelengthofeachsegmentusingsegmentadditionpostulate.Drawadiagramtorepresentthesituation.a.Point𝒁 isbetweenpoints𝑻 and𝑪.Thepointsarecollinear. 𝐓𝐙 = 𝟑𝒄𝒎𝒁𝑪 = 𝟏𝟐𝒄𝒎𝑻𝑪 =?

MeasuringSegments

SampleProblem3: Findthelengthofeachsegmentusingsegmentadditionpostulate.Drawadiagramtorepresentthesituation.a.Point𝒁 isbetweenpoints𝑻 and𝑪.Thepointsarecollinear. 𝐓𝐙 = 𝟑𝒄𝒎𝒁𝑪 = 𝟏𝟐𝒄𝒎𝑻𝑪 =?𝑻𝒁𝑪

𝟑𝒄𝒎𝟏𝟐𝒄𝒎

𝑻𝑪 = 𝑻𝒁 + 𝒁𝑪𝑻𝑪 = 𝟑𝒄𝒎 + 𝟏𝟐𝒄𝒎 = 𝟏𝟓𝒄𝒎𝑻𝑪 = 𝟏𝟓𝒄𝒎

MeasuringSegments

SampleProblem3: Findthelengthofeachsegmentusingsegmentadditionpostulate.Drawadiagramtorepresentthesituation.b.Point𝑺 isbetweenpoints𝑸 and𝑲.Thepointsarecollinear. 𝑸𝑲 = 𝟐𝟒𝒄𝒎𝑸𝑪 = 𝟏𝟎𝒄𝒎𝑺𝑲=?

MeasuringSegments

SampleProblem3: Findthelengthofeachsegmentusingsegmentadditionpostulate.Drawadiagramtorepresentthesituation.b.Point𝑺 isbetweenpoints𝑸 and𝑲.Thepointsarecollinear.𝑸𝑲 = 𝟐𝟒𝒄𝒎𝑸𝑪 = 𝟏𝟎𝒄𝒎𝑺𝑲=?

𝑸𝑺𝑲𝟐𝟒𝒄𝒎

𝟏𝟎𝒄𝒎

𝑸𝑲 = 𝑸𝑺 + 𝑺𝑲𝑺𝑲 = 𝑸𝑲−𝑸𝑺𝑺𝑲 = 𝟐𝟒𝒄𝒎 − 𝟏𝟎𝒄𝒎 = 𝟏𝟒𝒄𝒎𝑺𝑲 = 𝟏𝟒𝒄𝒎

MeasuringSegments

• Themidpoint isthemiddlepointofalinesegment.Itisequidistantfrombothendpoints,anditisthecentroidbothofthesegmentandoftheendpoints.• Themidpointofasegment,dividesthesegmentintotwocongruentsegments.• Asegmentbisector isapoint,lineoralinesegmentthatpassesthroughamidpointofanothersegment.• Asegmentbisectorisusuallyatthecenterofthelinesegmentandalwaysincludesthemidpointofthelinesegment.

MeasuringSegments

𝒉

𝒉isasegmentbisectorof𝑨𝑪𝑩isamidpointof𝑨𝑪

𝑨𝑩𝑪

MeasuringSegments

SampleProblem4: Findthelengthofeachsegment.Drawadiagramtorepresentthesituation.a. Point𝑩 ismidpointofsegment𝑴𝑵.𝑴𝑵 = 𝟏𝟎𝒄𝒎𝑴𝑩 =?𝑩𝑵 =?

MeasuringSegments

SampleProblem4: Findthelengthofeachsegment.Drawadiagramtorepresentthesituation.a. Point𝑩 ismidpointofsegment𝑴𝑵.𝑴𝑵 = 𝟏𝟎𝒄𝒎𝑴𝑩 =?𝑩𝑵 =?

𝑴𝑩𝑵

𝟏𝟎𝒄𝒎

MeasuringSegments

SampleProblem4: Findthelengthofeachsegment.Drawadiagramtorepresentthesituation.a. Point𝑩 ismidpointofsegment𝑴𝑵.𝑴𝑵 = 𝟏𝟎𝒄𝒎𝑴𝑩 =?𝑩𝑵 =?

𝑴𝑵 = 𝑴𝑩 + 𝑩𝑵𝑴𝑩 = 𝑩𝑵𝑴𝑵 = 𝟐 ∗𝑴𝑩

𝑴𝑩 =𝑴𝑵𝟐 =

𝟏𝟎𝒄𝒎𝟐 = 𝟓𝒄𝒎 𝑩𝑵 = 𝟓𝒄𝒎

𝑴𝑩 = 𝑩𝑵

MeasuringSegments

SampleProblem4: Findthelengthofeachsegment.Drawadiagramtorepresentthesituation.b. Point𝑫 ismidpointofsegment𝑭𝑻.𝑭𝑫 = 𝟔𝒊𝒏.𝑫𝑻 =?𝑭𝑻 =?

MeasuringSegments

SampleProblem4: Findthelengthofeachsegment.Drawadiagramtorepresentthesituation.b. Point𝑫 ismidpointofsegment𝑭𝑻.𝑭𝑫 = 𝟔𝒊𝒏.𝑫𝑻 =?𝑭𝑻 =?

𝑭𝑫𝑻

𝟔𝒊𝒏.

MeasuringSegments

SampleProblem4: Findthelengthofeachsegment.Drawadiagramtorepresentthesituation.b. Point𝑫 ismidpointofsegment𝑭𝑻.𝑭𝑫 = 𝟔𝒊𝒏.𝑫𝑻 =?𝑭𝑻 =?

𝑭𝑫 = 𝑫𝑻𝑫𝑻 = 𝟔𝒊𝒏.

𝑭𝑻 = 𝑭𝑫 + 𝑫𝑻𝑭𝑻 = 𝟔𝒊𝒏. +𝟔𝒊𝒏. = 𝟏𝟐𝒊𝒏.𝑭𝑻 = 𝟏𝟐𝒊𝒏.

MeasuringSegments

SampleProblem5: Findthecoordinateofthemidpointofeachsegment.a.

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𝑹𝑩

MeasuringSegments

SampleProblem5: Findthecoordinateofthemidpointofeachsegment.a.

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𝑹𝑩

𝑴𝑹𝑩 =−𝟐 + 𝟐𝟐 =

𝟎𝟐

𝑴𝑹𝑩 = 𝟎 Thecoordinateofthemidpointof 𝑹𝑩is𝟎

MeasuringSegments

SampleProblem5: Findthecoordinateofthemidpointofeachsegment.b.

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𝑲𝑳

MeasuringSegments

SampleProblem5: Findthecoordinateofthemidpointofeachsegment.b.

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𝑲𝑳

𝑴𝑲𝑳 =−𝟑 + 𝟏𝟐 =

−𝟐𝟐

𝑴𝑲𝑳 = −𝟏Thecoordinateofthemidpointof𝑲𝑳is − 𝟏

MeasuringSegments

• Asegmentpartition isapoint,lineoralinesegmentthatpartitionsthelinesegmentinaparticularratio.• Iftheratioisequal,thesegmentpartitionbecomesasegmentbisector.

MeasuringSegments

SampleProblem6: Dividethelinesegmentintheratiogivenbyputtingadotatthepartition.a.

𝑽𝑩

𝑹𝒂𝒕𝒊𝒐𝟏: 𝟒

MeasuringSegments

SampleProblem6: Dividethelinesegmentintheratiogivenbyputtingadotatthepartition.a.

𝑽𝑩

𝑹𝒂𝒕𝒊𝒐𝟏: 𝟒

MeasuringSegments

SampleProblem6: Dividethelinesegmentintheratiogivenbyputtingadotatthepartition.b.

𝑯𝑳

𝑹𝒂𝒕𝒊𝒐𝟑: 𝟏

MeasuringSegments

SampleProblem6: Dividethelinesegmentintheratiogivenbyputtingadotatthepartition.b.

𝑯𝑳

𝑹𝒂𝒕𝒊𝒐𝟑: 𝟏