1. 4.1 Detours and Midpoints
Objectives: To use detours in proofs and to apply the midpoint formula.

2. Procedure for Detour Proofs
Determine which triangles must be congruent to reach the required conclusion.
Attempt to prove that these triangles are congruent. If you don’t have enough information to prove them congruent, take a DETOUR (follow steps 3 – 5).
Determine which parts are necessary in order to prove that the triangles are congruent.
4. FIND a pair of triangles
(a) that you can easily prove to be congruent
(b) that contain the parts needed for the main proof
Prove the triangles found in Step 4 congruent
6. Use CPCTC and complete the proof

3. EX 1 Given: AB ≅ AD BC ≅ CD Prove: ΔABE ≅ ΔADE
Statements Reasons
AB ≅ AD Given
BC ≅ CD Given
AC ≅ AC Reflexive
ΔABC ≅ ΔADC 4. SSS
BAE ≅ DAE 5. CPCTC
AE ≅ AE Reflexive
ΔABE ≅ ΔADE 7. SAS

4. Theorem
If A = (x1, y1) and B = (x2, y2), then the midpoint of AB can be found by using the midpoint formula

5. EX 2 In ΔXYZ find the coordinates of the point at which the median from X intersects YZ.
A median is drawn from a vertex
to the midpoint of the opposite
side. Use the midpoint formula
to find the coordinates of point M.

6. Partner Proofs
Work with your partner to complete #1 – 4.

7. Homework
p. 173 #2 – 6
HL Practice Problems #1-4