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Mastering Detours and Midpoints in Geometric Proofs

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This lesson teaches you how to effectively use detours in proofs and apply the midpoint formula. You will learn the step-by-step process of determining which triangles need to be proven congruent and how to take detours when information is insufficient. The lesson includes practical examples, such as finding midpoints on a number line and applying the midpoint formula in coordinate geometry. By the end of this lesson, you will confidently navigate geometric proofs and understand how to establish congruence through detours.

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Mastering Detours and Midpoints in Geometric Proofs

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  1. 4.1 Detours and Midpoints Objective: After studying this lesson you will be able to use detours in proofs and apply the midpoint formula.

  2. A Given: B D E Conclusion: C Statement Reason 1. Given 2. Given 3. Reflexive Property 4. SSS (1,2,3) 5. CPCTC 6. Reflexive Property 7. SAS (1,5,6)

  3. 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.

  4. 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.)

  5. 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.

  6. 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 14 A M 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

  7. 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:

  8. Given: P Z Y Conclusion: W Q X

  9. Find the coordinates of M, the midpoint of segment AB B (7, 6) A (-1, 3)

  10. 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)

  11. Summary: Describe what you will do if there is not enough information to prove with the given information. Homework: worksheet

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