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1.3 Key Concepts

1.3 Key Concepts. Midpoint. The Point that divides the segment into two congruent segments. Segment Bisector. A Point, Ray, Line, Line Segment, or Plane that intersects the segment at its Midpoint. Midpoint Formula. [(x 1 + x 2 )/2] , [(y 1 + y 2 )/2]. Distance Formula.

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1.3 Key Concepts

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  1. 1.3 Key Concepts

  2. Midpoint The Point that divides the segment into two congruent segments.

  3. Segment Bisector A Point, Ray, Line, Line Segment, or Plane that intersects the segment at its Midpoint

  4. Midpoint Formula [(x1 +x2)/2], [(y1 + y2)/2]

  5. Distance Formula AB = √[(x2-x1)2 + (y2-y1)2]

  6. 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.

  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. In Exercises 1 and 2, identify the segment bisectorof PQ . Then find PQ. 1. ANSWER MN; 3 3 4 for Examples 1 and 2 GUIDED PRACTICE

  10. 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

  11. 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

  12. SOLUTION 1 , – , M M = 2 5 a.FIND MIDPOINTUse the Midpoint Formula. 2 The coordinates of the midpoint Mare 1 5 – , 2 2 ANSWER – 3 + 2 1 + 4 2 2 EXAMPLE 3 Use the Midpoint Formula

  13. b.FIND ENDPOINTThe midpoint of JKis M(2, 1). One endpoint is J(1, 4). Find the coordinates of endpoint K. EXAMPLE 3 Use the Midpoint Formula

  14. STEP 1 Find x. STEP 2 Find y. 4+ y 1+ x 1 2 = = 2 2 ANSWER The coordinates of endpoint Kare (3, – 2). EXAMPLE 3 Use the Midpoint Formula SOLUTION FIND ENDPOINTLet (x, y) be the coordinates of endpoint K. Use the Midpoint Formula. 4 + y = 2 1 + x = 4 y =–2 x =3

  15. 3. The endpoints of ABare A(1, 2) andB(7, 8). Find the coordinates of the midpoint M. ANSWER (4,5) 4. The midpoint of VWis M(– 1, – 2). One endpoint is W(4, 4). Find the coordinates of endpoint V. ANSWER (– 6, – 8) for Example 3 GUIDED PRACTICE

  16. Use the Distance Formula. You may find it helpful to draw a diagram. EXAMPLE 4 Standardized Test Practice SOLUTION

  17. RS = ~ = 2 2 = (x– x) + (y–y) 2 1 2 1 = 2 2 [(4 – 2)] + [(–1) –3] = 2 2 (2) + (–4 ) = 4+16 4.47 20 ANSWER The correct answer is C. EXAMPLE 4 Standardized Test Practice Distance Formula Substitute. Subtract. Evaluate powers. Add. Use a calculator to approximate the square root.

  18. 6. What is the approximate length of AB, with endpoints A(–3, 2) and B(1, –4)? ANSWER B 6.1 units 7.2 units 8.5 units 10.0 units for Example 4 GUIDED PRACTICE

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