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10.1 Distance and Midpoint Formulas
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10.1 Distance and Midpoint Formulas

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  • 1. 10.1 The Distance and Midpoint Formulas Algebra 2 Mr. Swartz
  • 2. Objectives/Standard/Assignment Objectives: 1. Find the distance between two points in the coordinate plane, and 2. Find the midpoint of a line segment in the coordinate plane.
  • 3. Geometry Review! • What is the differenceWhat is the difference between the symbols AB andbetween the symbols AB and AB?AB? Segment ABSegment AB TheThe lengthlength ofof Segment ABSegment AB
  • 4. The Distance Formula • The Distance d between the points (x1,y1) and (x2,y2) is : 2 12 2 12 )()( yyxxd −+−=
  • 5. The Pythagorean Theorem states that if a right triangle has legs of lengths a and b and a hypotenuse of length c, then a2 + b2 = c2 . Remember!
  • 6. Find the distance between the two points. • (-2,5) and (3,-1)(-2,5) and (3,-1) • Let (xLet (x11,y,y11) = (-2,5) and (x) = (-2,5) and (x22,y,y22) = (3,-1)) = (3,-1) 22 )51())2(3( −−+−−=d 3625+=d 81.761 ≈=d
  • 7. Classify the Triangle using theClassify the Triangle using the distance formula (as scalene,distance formula (as scalene, isosceles or equilateral)isosceles or equilateral) 29)61()46( 22 =−+−=AB 29)13()61( 22 =−+−=BC 23)63()41( 22 =−+−=AC Because AB=BC the triangle isBecause AB=BC the triangle is ISOSCELESISOSCELES C: (1.00, 3.00) B: (6.00, 1.00) A: (4.00, 6.00) C B A
  • 8. The Midpoint Formula • The midpoint between the twoThe midpoint between the two points (xpoints (x11,y,y11) and (x) and (x22,y,y22) is:) is: ) 2 , 2 ( 1212 yyxx m ++ =
  • 9. MIDPOINT FORMULA
  • 10. Find the midpoint of theFind the midpoint of the segment whose endpointssegment whose endpoints are (6,-2) & (2,-9)are (6,-2) & (2,-9)       −+−+ 2 92 , 2 26       − 2 11 ,4
  • 11. Find the coordinates of the midpoint of GH with endpoints G(–4, 3) and H(6, –2). Substitute. Write the formula. Simplify. Additional Example 1: Finding the Coordinates of a Midpoint G(–4, 3) H(6, -2)
  • 12. Additional Example 2: Finding the Coordinates of an Endpoint Step 1 Let the coordinates of P equal (x, y). Step 2 Use the Midpoint Formula. P is the midpoint of NQ. N has coordinates (–5, 4), and P has coordinates (–1, 3). Find the coordinates of Q.
  • 13. Additional Example 2 Continued Multiply both sides by 2. Isolate the variables. –2 = –5 + x +5 +5 3 = x 6 = 4 + y −4 −4 Simplify. 2 = y Set the coordinates equal. Step 3 Find the x- coordinate. Find the y-coordinate. Simplify.