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Chapter 4.1 – Detours and Midpoints

Chapter 4.1 – Detours and Midpoints. Jason Kohn Evan Eckersley and Shomik Ghosh. 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. Detour Proofs.

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Chapter 4.1 – Detours and Midpoints

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  1. Chapter 4.1 – Detours and Midpoints Jason Kohn Evan Eckersley and Shomik Ghosh

  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

  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)

  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)

  5. This concludes the concepts of detour proofs. • Now, let’s 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.

  6. 1. Given: ΔSHK is isosceles with base HK HI ≅ KO SM bisects ∠ OSI Prove: OM ≅ MI STATEMENTS REASONS 1.) ΔSHK is isos. with base HK 2.) HI ≅ KO 3.) SM bisects ∠ OSI 4.) ∠H ≅ ∠K 5.) HS ≅ KS 6.) HO ≅ KI 7.) ΔSHO ≅ ΔSKI 8.) SO ≅ SI 9.) ∠OSM ≅ ∠ISM 10.) SM ≅ SM 11.) ΔOSM ≅ ΔISM 12.) OM ≅ MI 1.) Given 2.) G 3.) G 4.) If a Δ is isos., then . 5.) If a Δ 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

  7. 2. Given: JA ≅ NK AO ≅ SN JS ≅ OK Prove: JN ≅ AK STATEMENTS REASONS 1.) JA ≅ NK 2.) AO ≅ SN 3.) JS ≅ OK 4.) AS ≅ ON 5.) Δ JAS ≅ Δ KNO 6.) ∠JSA ≅ ∠KON 7.) ∠JSN ≅ ∠ KOA 8.) Δ JSN ≅ ΔKOA 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

  8. 3. (Do it on your own!) Given: ∠ FTN ≅ ∠ SAN AN ≅ NT Prove: Δ FYS is isosceles STATEMENTS REASONS

  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.

  10. Midpoint Formula • 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…

  11. Theorem 22 • If A = (x1, y1) and B = (x2,y2), then the midpoint M = (xm, ym) of AB can be found by using the midpoint formula. (Rhoad 170-171)

  12. How does one do this? (4, 3) M B (-2, 1) A 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 2 2 4 2 (1, 2) , = , =

  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.

  14. 1. A circle has M has its center (Circle M) and AB as a diameter. Using the coordinates given, find M. (6, 6) A M B (4, 2) 4 + 6 2 2 + 6 2 10 2 8 2 , = , = (5, 2)

  15. O 2. E H 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 N P I W X

  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.

  17. Work Cited Rhoad, Richard, et al. Geometry for Enjoyment and Challenge. Boston: McDougal Littell, 1997. Print.

  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.

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