4.1 Detours & 4.1 Detours & MidpointsMidpoints

Test will be handed Test will be handed out after lesson.out after lesson.

Obj: Use detours in proofsObj: Use detours in proofs

Apply the midpoint Apply the midpoint formulas formulas

Warm UpWarm Up

Detour Proofs: used when Detour Proofs: used when you need to prove 2 pairs of you need to prove 2 pairs of

’s = to solve a case. ’s = to solve a case.

Ex:1Ex:1

A E A E Given: AB = ADGiven: AB = AD

BC = CDBC = CD

B D B D Prove: ABE =Prove: ABE = ADEADE

Do we have Do we have enoughenough

C C info?info?We only have sides AB = AD & AE = AEWe only have sides AB = AD & AE = AE

We need an angleWe need an angle

Prove ABC = ADC First Prove ABC = ADC First by SSSby SSS

StatementsStatements• (S) AB = AD(S) AB = AD• (S) BC = DC(S) BC = DC• (S) AC = AC(S) AC = AC• ABC = ADCABC = ADC• (A) BAC = ADC(A) BAC = ADC• (S) AE = AE(S) AE = AE• ABE = ADEABE = ADE

EX.1 cont.

ReasonsReasons• GivenGiven• GivenGiven• Reflexive PropertyReflexive Property• SSS (1,2,3)SSS (1,2,3)• CPCTCCPCTC• Reflexive Property Reflexive Property • SAS (1,5,6)SAS (1,5,6)

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Midpoint formula: for the Midpoint formula: for the midpoint of a line: midpoint of a line: X = X + X = X +

XX Take the average of two given pointsTake the average of two given points

EX.2: Find the midpoint of line segment ABEX.2: Find the midpoint of line segment AB

A X BA X B X = X = -2 -2

+ 8+ 8

= = 66

equal distance = equal distance = 33

hence hence midpointmidpoint

Midpoint formula for segment on the coordinate plane:

EX.3

Find mdpt. of segment (1, 4) (6, 2) (x, y) (x , y ) Ans:

m (xm, ym ) = 1+6 , 4+2

= 7 , 6 = 7 ,

The EndThe End