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Geometry Section1.3

Geometry Section1.3 . Using Segments and Congruence Distance and Midpoint Formula. M. P. Q. What is midpoint?. The midpoint M of PQ is the point between P and Q such that PM = MQ. How do you find the midpoint?

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Geometry Section1.3

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  1. Geometry Section1.3 Using Segments and Congruence Distance and Midpoint Formula

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

  3. -5 0 6 Examples: 1.) Find the midpoint of AC: (-5 + 6)/2 = ½ 2.) If M is the midpoint of AZ, AM = 3x + 12 and MZ = 6x – 9; find the measure of AM and MZ. 3x + 12 = 6x – 9 21 = 3x X = 7 AM = 33 MZ = 33

  4. 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)

  5. 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)

  6. 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

  7. 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.

  8. 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

  9. 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.

  10. 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.

  11. 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

  12. 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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