4.1 Detours & 4.1 Detours & MidpointsMidpoints

Obj: Use detours in proofsObj: Use detours in proofs

Apply the midpoint Apply the midpoint formulas formulas

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

s to solve a case.

Ex:1Ex:1

A E A E Given: AB Given: AB AD AD

BC BC CD CD

B D B D Prove: ABE Prove: ABE ADEADE

Do we have enough Do we have enough info?info?

We only have sides AB AD & AE AEWe need an angle.

CC

Prove ABC Prove ABC ADC ADC First by SSSFirst by SSS

StatementsStatements

1.1. (S) AB (S) AB AD AD

2.2. (S) BC (S) BC DC DC

3.3. (S) AC (S) AC AC AC

4.4. ABC ABC ADC ADC

5.5. (A) (A) BAC BAC DACDAC

6.6. (S) AE (S) AE AE AE

7.7. ABE ABE ADE ADE

EX.1 cont.

ReasonsReasons

1.1. GivenGiven

2.2. GivenGiven

3.3. Reflexive Reflexive PropertyProperty

4.4. SSS (1,2,3)SSS (1,2,3)

5.5. CPCTCCPCTC

6.6. Reflexive Reflexive Property Property

7.7. SAS (1,5,6)SAS (1,5,6)

Procedure for Detour Procedure for Detour ProofsProofs

1.1. Determine which triangles you Determine which triangles you must prove to be congruent to must prove to be congruent to reach the required conclusion.reach the required conclusion.

2.2. Attempt to prove that these Attempt to prove that these triangles are congruent. If you triangles are congruent. If you cannot do so for lack of enough cannot do so for lack of enough information, take a detour.information, take a detour.

3.3. Identify the parts that you must Identify the parts that you must prove to be congruent to prove to be congruent to establish the congruence of the establish the congruence of the triangles.triangles.

4.4. Find a pair of triangles thatFind a pair of triangles that1.1. You can readily prove to be You can readily prove to be

congruent.congruent.

2.2. Contain a pair of parts needed Contain a pair of parts needed for the main proof.for the main proof.

5.5. Prove that the triangles Prove that the triangles found in step 4 are found in step 4 are congruent.congruent.

6.6. Use CPCTC and complete the Use CPCTC and complete the proof planned in step 1.proof planned in step 1.

Procedure for Detour Procedure for Detour ProofsProofs

Midpoint formula: for the Midpoint formula: for the midpoint of a line take the midpoint of a line take the average of two given points. average of two given points. XXmm = X = X11 + X + X22

EX.2: Find the midpoint of line segment EX.2: Find the midpoint of line segment ABAB

equal distance, hence equal distance, hence midpointmidpoint

X = X = -2 + 8-2 + 8 2 2= = 66 2 2=3=3

22

A A B B-2-2 88

XX33

Midpoint formula for segment on the coordinate plane:

• Find the midpoint of (1, 4) and (6, 2).Find the midpoint of (1, 4) and (6, 2).• 1 + 61 + 6, , 4 + 24 + 2

2 2 2 2

• ((77//22, , 66//22))

• (3.5, 3)(3.5, 3)

(( ))