Chapter 4Lines in the Plane

Section 4.1 Detours and Midpoints

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

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 congruentContain a pair of parts needed for the

objective of the main proof Use CPCTC and complete the proof

Detour Proof Example

Given: 1 2 , 3 4

Prove: DE BE

Detour Proof Example

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

1 1 2 2

1 2 1 2

If ( , ) and ( , ), then the midpoint

( , ) of AB can be found by using the midpoint formula:

( , ) ,2 2

m m

m m

A x y B x y

M x y

x x y yM x y

1 2 1 2( , ) ,2 2m m

x x y yM x y

Visual Approach

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.

1 ( 8)

2, 6 ( 4)

2

7

2, 10

2

7

2, 5

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.

Consider triangle ABC where A=(1,1) B=( 5,2 ) C=( 4,6 )

1.Sketch the triangle.2.Show that the triangle is isosceles

using the distance formula.3.Algebra review of slopes…can you

show that this is a right isosceles triangle?