4.1 Detours and Midpoints Objectives: Use detours in proofs

1. 4.1 Detours and Midpoints Objectives: • Use detours in proofs • Find midpoints of segments using formula

2. New Assumption supplementary angles from diagram

3. Midpoint Formula: Distance Formula: Example 1: Find the midpoint and distance of the segment with endpoints: (4, 9) and (–2, 10).

4. Example 2: Find coordinates of A if B(–3,–1), M(3, –4) and M is the midpoint of .

5. Example 3: If R is the midpoint between Q and S, find the coordinates of Q if R(7, 1) and S(2, 8).

6. Tips to proofs: • 1. Determine which triangles you must prove to be congruent to reach the required conclusion. • 2. Attempt to prove that these triangles are congruent. If you cannot do so for lack of enough given information, take a detour. • 3. Identify the parts that you must prove to be congruent to establish the congruence of the triangles. • 4. Find a pair of triangles that: • I. You can prove congruent. • II. Contain the pairs of parts you need. • 5. Prove that the triangles are congruent. • 6. Use CPCTC to get that parts you need and finish the proof.

7. D C Example 4: E A B Given Given Reflexive Property SSS ∆ABC ∆ADC DAC BAC CPCTC Reflexive Property ∆ABE ∆ADE SAS

8. Example 5: R U T S Q Given Definition of bisect RT = ST Definition of congruent segments All radii of a circle are congruent Reflexive Property ∆RUT ∆SUT SSS RUT SUT CPCTC Reflexive Property Continued on next slide

9. Example 5: R U T S Q SAS ∆RUQ ∆SUQ CPCTC RQU  SQU Definition of congruent angles mRQU = mSQU Definition of bisect

10. R Q S Example 6: V T U Given Definition of bisect QR = RS Definition of congruent segments Given Q S Given ∆RQV ∆RST SAS CPCTC Given Continued on next slide

11. R Q S Example 6: V T U Definition of midpoint VU = UT Definition of congruent segments Reflexive Property SSS ∆RVU ∆RTU VUR TUR CPCTC