1. Bellwork Find the average of -11 & 5 Solve Simplify Find to the nearest hundredth Clickers

2. Bellwork Solution Find the average of -11 & 5

3. Bellwork Solution Solve

4. Bellwork Solution Simplify

5. Bellwork Solution Find to the nearest hundredth

6. Segments and Congruence & Use Midpoint and Distance Formulas Sections 1.2 &1.3

7. The Concept Today we’re going to start with the idea of congruence and continue onto two monumental yet simple postulates We will then see the definition for a midpoint and the formula for finding it We’ll also learn the formula for finding the length of a line via the distance formula

8. Definitions Postulate –Rule that is accepted without proof –Can also be called an axiom Vertex Axis of symmetry Postulate1.2 Rule that is accepted without proof

9. Ruler postulate This postulate explains that two points on a line can be explained as two coordinates –The distance between the two coordinates correlates to the length of the segment AB x1x1 x2x2

10. Ruler Postulate uses One use is if we assign values to the coordinates AB x1x1 x2x2 (2,0)(12,0) Or it can be used to allow the use of a ruler to find the length of a segment –Seems nonsensical, but again is something that must be explained in order to base further exploration

11. Segment Addition Postulate This postulate explains that two connected segments created by a point between two others can be added together to get the full distance It can also be used to explain interior points –If B is between A & C then AB+BC=AC –If AB+BC=AC, then B is between A and C ACB

12. Segment Addition Postulate use Based on this postulate find the length of BC, if AC=32 ACB 13

13. Congruence Congruence –The same measure as –AB is congruent to CD –Written as –Important that congruence is used in lieu of equals

14. Nomenclature At this point we should also discuss the two different nomenclatures you may see regarding segments This denotes the segment AB This denotes the length of segment AB

15. Use of Congruence Given the following points are XY and WZ congruent? –X: (-2, -5) –Y: (-2, 3) –W: (-4,3) –Z: (4, 3)

16. Definitions Midpoint –The point on a line segment that lies exactly halfway between the two endpoints –Divides the segment into two congruent pieces Vertex Axis of symmetry Midpoint1.3 The point on a line segment that lies exactly halfway between the two endpoints Divides the segment into two congruent pieces A B C

17. Finding a midpoint How do we find a midpoint? –We simply divide the length by two –AB is 20 –What is AC? ABC

18. Example If point X is the midpoint of segment JK and the length of JX is 14.5, what length of segment JK? JKX

19. Bisectors If a line, ray or segment goes through the midpoint of another segment, it is called the bisector of the segment AB C D E

20. Showing congruence We are able to show congruence of segments in a figure through the use of slash marks Using the same diagram, in which segment AB is bisected AB C D E

21. Midpoints What happens if we put a line on the coordinate plane? –How do we find the midpoint? We can use a derivation of the ruler principle to find the midpoint of a line on the coordinate axis… –The formula is (x 1,y 1 ) (x 2,y 2 ) B

22. Where does this come from? How did we get this formula? B (x 1,y 1 ) ½(x 1 +x 2 ) (x 2,y 2 ) x2x2 ½ (y 1 +y 2 ) y2y2

23. Click-In What is the midpoint of a line segment that goes from (1, 2) to (11,20)?

24. Long Distance Call In addition to finding the midpoint of a line when on the coordinate plane, we can also find the distance or length of the segment using the ruler postulate and the pythagorean theorem The ruler postulate gives us length, but only in one dimension The Pythagorean Theorem gives us the length of the hypotenuse of a triangle if we have the length of the two sides So…we use the ruler postulate to figure the two lengths and then apply the Pythagorean theorem Let’s take a look…

25. Where does this come from? How did we get this formula? (x 1,y 1 ) (x 2,y 2 ) x 2 -x 1 y 2 -y 1 a= b= c Why is there not a plus or minus in front of this?

26. Example Find the distance between (-10,2) & (4,1)

27. On your own Find the distance between (-1,-1) & (10,2)

28. On your own Find the distance between (3,3) & (-2,-2)

29. Homework 1.2 –1,6-12 even, 16-30 1.3 –2-8, 20-22, 28-34 even, 43-45, 49

30. Practical Example

31. Most Important Points Definition of Congruence Segment addition postulate Definition of Midpoint Definition of Bisector Showing congruency in segments Midpoint Formula Distance Formula