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Bellwork

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

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Bellwork

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

  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 Rule that is accepted without proof Postulate 1.2 Axis of symmetry Vertex

  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 A B x1 x2

  10. Ruler Postulate uses • One use is if we assign values to the coordinates A B x1 (2,0) (12,0) x2 • 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 A B C

  12. Segment Addition Postulate use • Based on this postulate find the length of BC, if AC=32 A B C 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 • The point on a line segment that lies exactly halfway between the two endpoints • Divides the segment into two congruent pieces C B Midpoint 1.3 A Axis of symmetry Vertex

  17. Finding a midpoint • How do we find a midpoint? • We simply divide the length by two • AB is 20 • What is AC? A C B

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

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

  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 D A B E C

  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 (x2,y2) B (x1,y1)

  22. Where does this come from? • How did we get this formula? (x2,y2) B y2 (x1,y1) ½ (y1+y2) ½(x1+x2) x2

  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? (x2,y2) c b= y2-y1 (x1,y1) a= x2-x1 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

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