1. Chapter 4 Lines in the Plane
Section 4.1
Detours and Midpoints

2. Detour Proofs
To solve some problems, it is necessary to prove an intermediate congruence on the way to achieving the objective of the proof
We call these detour proofs

3. Procedure for Detour Proofs
Determine which triangles you must prove to be congruent to reach the required conclusion
Attempt to prove that these triangles are congruent. If you cannot do so for lack of enough given information, take a detour…
Detour: Find a pair of triangles that:
You can readily prove to be congruent
Contain a pair of parts needed for the objective of the main proof
Use CPCTC and complete the proof

4. Detour Proof Example

5. Detour Proof Example

6. The Midpoint Formula
We can apply the averaging process to develop a formula, called the midpoint formula, that can be used to find the coordinates of the midpoint of any segment in the coordinate plane

7. Visual Approach

8. Example
Find the midpoint of the line segment with endpoints (1, -6) and (-8, -4).
Solution To find the coordinates of the midpoint, we average the coordinates of the endpoints.

9. Examples: Find the midpoint of segment AB if A:(-3, 7) and B:(9, -20)
(3, -1) is the midpoint of segment RS.
If R = (12, -5) find the coordinates of S.

10. Consider triangle ABC where A=(1,1) B=( 5,2 ) C=( 4,6 )
Sketch the triangle.
Show that the triangle is isosceles using the distance formula.
Algebra review of slopes…can you show that this is a right isosceles triangle?