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Section 2.1

Learn about the properties of segments, measure of segments, segment addition postulate, midpoint formula, and endpoint calculations.

lburrows
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Section 2.1

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  1. Section 2.1 Using Segments and Congruence Midpoint Formula

  2. Objectives – What we’ll learn… • Apply the properties of real numbers to the measure of segments.

  3. A B D C Segments Where is B located? Between A and C Where is D located? Not between A and C For a point to be between two other points, all three points must be collinear. Segments can be defined using the idea of betweenness of points.

  4. Measure of Segments C B A What is a segment? A part of a line that consists of two endpoints and all the points between them. What is the measure of a segment? The distance between the two endpoints. In the above figure name three segments: CB BA AC

  5. Postulate 2-1Ruler Postulate The distance between points A and B, written as AB, is the absolute value of the difference of the coordinates of A and B. Use |Absolute Value|!!! X Y Since x is at -2 and Y is at 4, we can say the distance from X to Y or Y to X is: -2 – 4 = 6 or 4 – (-2) = 6

  6. Measure the length of STto the nearest tenth of a centimeter. Align one mark of a metric ruler with S. Then estimate the coordinate of T. For example, if you align Swith 2, T appears to align with 5.4. ST=5.4 – 2 = 3.4 The length of STis about 3.4 centimeters. ANSWER EXAMPLE 1 Apply the Ruler Postulate SOLUTION Use Ruler Postulate.

  7. Summary What do we use to find the distance between two points? |Absolute Value|

  8. Section 2.2 Using Segments and Congruence Distance and Midpoint Formula

  9. Postulate 2-2 Segment Addition Postulate If Q is between P and R, then PQ + QR = PR. If PQ +QR = PR, then Q is between P and R. 2x 4x + 6 R P Q PQ = 2x QR = 4x + 6 PR = 60 Use the Segment Addition Postulate find the measure of PQ and QR.

  10. Step 1: PQ + QR = PR (Segment Addition) 2x + 4x + 6 = 60 6x + 6 = 60 6x = 54 x =9 PQ = 2x = 2(9) = 18 QR =4x + 6 = 4(9) + 6 = 42 Step 2: Step 3: Step 4:

  11. Steps • Draw and label the Line Segment. • Set up the Segment Addition/Congruence Postulate. • Set up/Solve equation. • Calculate each of the line segments.

  12. Use the diagram to find GH. FH = FG+ GH 36 21+GH = 15 GH = EXAMPLE 3 Find a length SOLUTION Use the Segment Addition Postulate to write an equation. Then solve the equation to find GH. Segment Addition Postulate. Substitute 36 for FHand 21 for FG. Subtract 21 from each side.

  13. Plot J(– 3, 4), K(2, 4), L(1, 3), and M(1, – 2) in a coordinate plane. Then determine whether JKand LMare congruent. To find the length of a horizontal segment, find the absolute value of the difference of the x-coordinates of the endpoints. JK = 2 – (– 3) = 5 EXAMPLE 4 Compare segments for congruence SOLUTION Use Ruler Postulate.

  14. LM = – 2 – 3 = 5 ~ JKand LMhave the same length. So, JK LM. Remember when we speak of length the bar does not go over the letters but it does when we speak of congruence. = ANSWER EXAMPLE 4 Compare segments for congruence To find the length of a vertical segment, find the absolute value of the difference of the y-coordinates of the endpoints. Use Ruler Postulate.

  15. Section 2.5 Midpoint Formula: Finding the midpoint and endpoint.

  16. M P Q What is midpoint? The midpoint M of PQ is the point between P and Q such that PM = MQ. Endpoint: P Endpoint: Q Midpoint: M

  17. How do you find the midpoint? • On a number line, the coordinate of the midpoint of a segment whose endpoints have coordinates a and b is (a + b)/2. Find the AVERAGE!

  18. -5 0 6 Examples: 1.) Find the midpoint of AC: Endpoint: -5 Endpoint: 6 (Finding Average of two numbers) (-5 + 6)/2 Midpoint: 1/2

  19. 2.) If M is the midpoint of AZ, AM = 3x + 12 and MZ = 6x –9; find the measure of AM and MZ. AM = MZ (Def. of Midpoint) 3x + 12 = 6x – 9 21 = 3x X = 7 AM =3x + 12 =3(7) + 12 = 33 MZ = 6x – 9 = 6(7) – 9 = 33 Step 2: Step 3: Step 4:

  20. Steps of finding midpoint. Find midpoint of (-3, 7) and (8, -4). • Endpoint 1: ( -3, 7 ) Endpoint 2: ( 8 , -4 ) Midpoint: ( , ) (Average of x) (Average of y)

  21. Steps • Draw and label the Line Segment. • Set up the GEOMETRY Expression. a) Segment Addition Postulate b) Definition of Midpoint c) Definition of Congruence 3. Set up/Solve equation. 4. Calculate each of the line segments.

  22. Steps of finding Endpoint! Find the other endpoint with endpoint (-5, 6) & midpoint (3/2, 5). • Endpoint 1: ( -5, 6 ) Endpoint 2: ( x , y ) Midpoint: ( , ) (8, 4) Solve Equations:

  23. Steps of finding Midpoint: • Write down the order pair. • Find the AVERAGE of the x1 and x2. (x1 + x2)/2 = • Find the AVERAGE of the y1 and y2. (y1 + y2) /2 = • Write them as an order pair.

  24. Example: 1.) Find the midpoint, M, of A(2, 8) and B(4, -4). x = (2 + 4) ÷ 2 = 3 y = (8 + (-4)) ÷ 2 = 2 M = (3, 2) 2.) Find M if N(1, 3) is the midpoint of MP where the coordinates of P are (3, 6). M = (-1, 0) Find AVERAGE of x -> <-Find AVERAGE of y

  25. Q. How do you find the midpoint of 2 ordered pairs? A. In a coordinate plane, the coordinates of the midpoint of a segment whose endpoints have coordinates (x1, y1) and (x2, y2) are ((x1 + x2)/2), (y1 + y2)/2)

  26. a.FIND MIDPOINTThe endpoints ofRSare R(1,–3) and S(4, 2). Find the coordinates of the midpoint M. EXAMPLE 3 Use the Midpoint Formula

  27. ALGEBRA Point Mis the midpoint of VW. Find the length of VM . STEP 1 Write and solve an equation. Use the fact that VM = MW. EXAMPLE 2 Use algebra with segment lengths SOLUTION VM= MW Write equation. 4x–1= 3x + 3 Substitute. x – 1 = 3 Subtract 3xfrom each side. x = 4 Add 1 to each side.

  28. STEP 2 Evaluate the expression for VMwhen x =4. So, the length of VMis 15. Check: Because VM = MW, the length of MWshould be 15. If you evaluate the expression for MW, you should find that MW = 15. MW = 3x + 3 = 3(4) +3 = 15 EXAMPLE 2 Use algebra with segment lengths VM = 4x – 1 = 4(4) – 1 = 15

  29. M A B C N Bisectors What is a segment bisector? - Any segment, line, or plane that intersects a segment at its midpoint. If B is the midpoint of AC, then MN bisects AC.

  30. Skateboard In the skateboard design, VWbisects XYat point T, and XT=39.9cm. Find XY. Point Tis the midpoint of XY . So, XT = TY = 39.9cm. EXAMPLE 1 Find segment lengths SOLUTION XY = XT + TY Segment Addition Postulate = 39.9 + 39.9 Substitute. = 79.8cm Add.

  31. In Exercises 1 and 2, identify the segment bisectorof PQ . Then find PQ. 2. 5 ANSWER line l ; 11 7 for Examples 1 and 2 GUIDED PRACTICE

  32. Distance Formula • The Distance Formula was developed from the Pythagorean Theorem Where d = distance x =x-coordinate and y=y-coordinate

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