Download 3.1 What are congruent figures?

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 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. (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
14
B
M
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
Z
P
Y
Conclusion: 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