Download 4.1 Detours and Midpoints

4.1 Detours and Midpoints
Objective:
After studying this lesson you will be able to
use detours in proofs and apply the midpoint
formula.
A
Given:
AB  AD
BC  CD
B
E
Conclusion: ABE  ADE
C
Statement
1. AB  AD
2. BC  CD
3. AC  AC
4. ABC  ADC
5. BAE  DAE
6. AE  AE
7. ABE  ADE
Reason
1.
2.
3.
4.
Given
Given
Reflexive Property
SSS (1,2,3)
5. CPCTC
6. Reflexive Property
7. SAS (1,5,6)
D
On the previous proof the only information that was
useful was that AB  AD . There did not seem to be
enough information to prove
ABE  ADE
We had to prove something else congruent first.
Proving something else congruent first is called
taking a little detour in order to pick up the
congruent parts that we need.
If you need a detour use the following procedure.
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 can’t, take a detour.
3. Find the parts that you must prove to be congruent
to prove congruent triangles.
(Remember that there are many ways to prove triangles
congruent. Consider them all.)
4. Find a pair of triangles that
a. You can readily prove to be congruent
b. Contain a pair of parts needed for the main proof
(parts identified in step 3).
5. Prove that the triangles found in step 4 are congruent.
6. Use CPCTC and complete the proof planned in step 1.
Midpoint
Example: On the number line below, the coordinate of A
is 2 and the coordinate of B is 14. Find the coordinate
of M, the midpoint of segment AB.
2
A
M
14
B
There are several ways to solve this problem.
One of these ways is the averaging process. We
add the two numbers and divide by 2.
The midpoint is 8
Theorem
If A = (x1, y1) and B = (x2, y2), then
the midpoint M = (xm, ym ) of
segment AB can be found by
using the midpoint formula:
 x1  x2 y1  y2 
M   xm , ym   
,

2 
 2
Given:
PQ bisects YZ
Q is the midpt. of WX
Y  Z , WZ  XY
Conclusion:
Z
P
Y
WQP  XQP
W
Q
X
Find the coordinates of M, the midpoint of segment
AB
B (7, 6)
A (-1, 3)
In triangle ABC, find the coordinates of the point at
which the median from A intersects BC
C (6, 10)
M
A (14, 5)
B (2, 4)
Summary:
Describe what you will do if
there is not enough
information to prove with
the given information.
Homework: worksheet