Objective: After studying this lesson you will be able to use detours in proofs and apply the midpoint

formula.

4.1 Detours and Midpoints

C

DGiven:

Conclusion:

E

A

B

Statement Reason1. Given2. Given3. Reflexive Property4. SSS (1,2,3)5. CPCTC

ADABCDBC

ADEABE

ADAB .1CDBC .2ACAC .3

ADCABC .4DAEBAE .5

AEAE .6 6. Reflexive Property7. SAS (1,5,6)ADEABE .7

On the previous proof the only information that was useful was that . There did not seem to be enough information to prove

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.

ADAB

ADEABE

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 thata. You can readily prove to be congruentb. 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.

MidpointExample: 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.

A B2 14

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:

1 2 1 2, ,2 2m m

x x y yM x y

Given:

Conclusion:

bisects PQ YZ�

Q is the midpt. of WXY , Z WZ XY

WQP XQP

W

Z

Q X

YP

Find the coordinates of M, the midpoint of segment AB

A (-1, 3)

B (7, 6)

In triangle ABC, find the coordinates of the point at which the median from A intersects BC

A (14, 5)

B (2, 4)

C (6, 10)

M

Summary:

Describe what you will do if there is not enough information to prove with the given information.

Homework: worksheet