Transcript of Jason Kohn Evan Eckersley and Shomik Ghosh Chapter 4.1 – Detours and Midpoints.
- Slide 1
- Jason Kohn Evan Eckersley and Shomik Ghosh Chapter 4.1 Detours
and Midpoints
- Slide 2
- Learning Goals Upon completing this Powerpoint, you will be
able to Understand detours, when to use them, and how to use them
Understand and apply the midpoint formula
- Slide 3
- Detour Proofs In some problems, there is not enough information
to prove a pair of triangles congruent. We must first prove a
different pair of triangles congruent by taking a little detour.
One can then use CPCTC to prove another pair of triangles
congruent. Whenever such a detour is taken to prove a statement,
the proof is referred to as a detour proof. (Rhoad 169)
- Slide 4
- Procedure for Detour Proofs 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 (steps
3-5 below). 3. Identify the parts that you must prove to be
congruent to establish the congruence if the triangles. 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. (Rhoad 170)
- Slide 5
- This concludes the concepts of detour proofs. Now, lets try our
hand at some sample problems. The first two sample problems are
designed to work with you step-by-step. The third sample problem is
for you to solve independently.
- Slide 6
- 1.) SHK is isos. with base HK 2.) HI KO 3.) SM bisects OSI 4.)
H K 5.) HS KS 6.) HO KI 7.) SHO SKI 8.) SO SI 9.) OSM ISM 10.) SM
SM 11.) OSM ISM 12.) OM MI 1.) Given 2.) G 3.) G 4.) If a is isos.,
then. 5.) If a is isos., then. 6.) Subtraction 7.) SAS (4, 5, 6)
8.) CPCTC 9.) If a bisects an , it divides the into two s. 10.)
Reflexive Property 11.) SAS (8, 9, 10) 12.) CPCTC STATEMENTSREASONS
Given: SHK is isosceles with base HK HI KO SM bisects OSI Prove: OM
MI 1.
- Slide 7
- Given: JA NK AO SN JS OK Prove: JN AK STATEMENTSREASONS 1.) JA
NK 2.) AO SN 3.) JS OK 4.) AS ON 5.) JAS KNO 6.) JSA KON 7.) JSN
KOA 8.) JSN KOA 9.) JN AK 1.) Given 2.) G 3.) G 4.) Subtraction 5.)
SSS (1, 3, 4) 6.) CPCTC 7.) Supps. of s are . 8.) SAS (2, 3, 7) 9.)
CPCTC 2.
- Slide 8
- STATEMENTSREASONS (Do it on your own!) Given: FTN SAN AN NT
Prove: FYS is isosceles 3.
- Slide 9
- This concludes detour proofs. By this point, you should have
mastered the concepts and usage of detour proofs. If not, please
refer to Chapter 4.1 of your math textbook for additional
assistance.
- Slide 10
- The midpoint formula can be used to locate the midpoint of a
segment. A midpoint is the bisection point of the segment.
Therefore, if the endpoints of the segment are given, you can
locate the midpoint of said segment. The midpoint formula is This
is derived from Midpoint Formula
- Slide 11
- If A = (x 1, y 1 ) and B = (x 2, y 2 ), then the midpoint M =
(x m, y m ) of AB can be found by using the midpoint formula.
Theorem 22 (Rhoad 170-171)
- Slide 12
- How does one do this? A M B (-2, 1) (4, 3) Segment AB is on a
coordinate plane (not shown in the diagram), and the coordinates of
the endpoints are given. Use the midpoint formula to find the
coordinates of point M. 4+(-2) 2, 3+1 2 = 2222, 4242 = (1, 2)
- Slide 13
- We now know how to determine the midpoints of a segment. Now
let us use Theorem 22 in a few sample problems (with imaginary
coordinate planes). The first sample problem is designed to work
with you step by step. The second one is for you to solve
independently. If needed, feel free to use a calculator.
- Slide 14
- A B M A circle has M has its center (Circle M) and AB as a
diameter. Using the coordinates given, find M. (6, 6) (4, 2) 4 + 6
2, 2 + 6 2 = 10 2, 8 2 =(5, 2) 1.
- Slide 15
- P H O E N I X W Given: P = (1, 7) O = (10, 11) N = (14, 5) X =
(2, -1) H is the midpoint of PO E is the midpoint of ON I is the
midpoint of NX W is the midpoint of XP Find the coordinates of H,
E, W and I 2.
- Slide 16
- This concludes midpoints. By this point, you should have
mastered the concepts and usage of the midpoint formula. If not,
please refer to Chapter 4.1 of your math textbook for additional
assistance.
- Slide 17
- Work Cited Rhoad, Richard, et al. Geometry for Enjoyment and
Challenge. Boston: McDougal Littell, 1997. Print.
- Slide 18
- Thank you for using this Powerpoint to help review for
midterms. Jason, Evan, and Shomik hoped that we helped you to the
best of our abilities. All diagrams were original, and any
resemblance to diagrams in the textbook is pure coincidence. Please
Enjoy This Video Made by Shomik, Evan, Jason, and our dear friend
Kyle Kocsis. In advance we would like to apologize for a couple
moments with hard to hear sound quality.
- Slide 19