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INSTRUCTORS Carleen Eaton Grant Fraser Eric Smith
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Lecture Comments (2)

1 answer

Last reply by: Dr Carleen Eaton
Sat Jul 27, 2013 10:25 AM

Post by Tami Cummins on July 18, 2013

Did you accidentally mix up the y's or are you using the coor
dinates of the third point you graphed?

Distance Formula

  • The Distance formula asserts that the distance between the points (x1, y1) and (x2, y2) in the coordinate plane is √(x2 – x1)2 + (y2 – y1)2. This formula is a special case of the Pythagorean theorem.
  • Make sure that you know and understand how to use this formula. It will be used in a lot of later work.
  • In the Distance formula, the order of the x coordinates does not matter. So take whichever difference is easier to compute. The same comment applies to the y coordinates.
  • After squaring the differences in the distance formula, be sure to take the positive square root of their sum.
  • In some problems, you are given the distance between two points and three of their coordinates. In this situation, use the Distance formula to find the fourth coordinate.

Distance Formula

Find the distance between the points ( − 2,1 ) and ( 7, − 8 )
  • d = √{( x2 − x1 )2 + ( y2 − y1 )2}
  • d = √{( − 7 − ( 2 ) )2 + ( − 8 − 1 )2}
  • d = √{ − 52 + ( − 9 )2}
  • d = √{25 + 81}
d = √{106}
Find the distance between the points ( 4,11 ) and ( − 2, − 6 )
  • d = √{( − 2 − 4 )2 + ( − 6 − 11 )2}
  • d = √{( − 6 )2 + ( − 17 )2}
  • d = √{36 + 289}
d = √{325} = 5√{13}
Find the distance between the points ( − 13, − 15 ) and ( − 9,10 )
  • d = √{( − 9 − ( − 13 ) )2 + ( 10 − ( − 15 ) )2}
  • d = √{42 + 252}
d = √{641}
Find x if the distance between the points ( 4, − 3 ) and ( x, − 7 ) is 6
  • 6 = √{( x − 4 )2 + [ ( − 7 ) − ( − 3 ) ]2}
  • 6 = √{( x − 4 )2 + ( − 4 )2}
  • 6 = √{( x − 4 )2 + 16}
  • 62 = ( x − 4 )2 + 16
  • 20 = ( x − 4 )2
  • ±√{20} = x − 4
x = 4 ±√{20}
Find y if the distance between the points ( − 5, − 2 ) and ( 3,y ) is 12
  • 12 = √{( 3 − ( − 5 ) )2 + ( y − ( − 2 ) )2}
  • 12 = √{82 + ( y + 2 )2}
  • 12 = √{64 + ( y + 2 )2}
  • 144 = 64 + ( y + 2 )2
  • 80 = ( y + 2 )2
  • ±√{80} = y + 2
y = − 2 ±√{80}
Find y if the distance between the points ( − 11, − 8 ) and ( − 2,y ) is 14
  • 14 = √{[ ( − 2 ) − ( − 11 ) ]2 + [ y − ( − 8 ) ]2}
  • 14 = √{81 + ( y + 8 )2}
  • 196 = 81 + ( y + 8 )2
  • 115 = ( y + 8 )2
  • ±√{115} = y + 8
y = − 8 ±√{115}
Find the distance between the points ( 2√3 ,7√6 ) and ( 9√3 , − 4√6 )
  • d = √{( 9√3 − 2√3 )2 + ( − 4√6 − 7√6 )2}
  • d = √{( 7√3 )2 + ( − 11√6 )2}
  • d = √{147 + 726}
d = √{873} = 3√{97}
Find the distance between the points ( − 11√5 ,4√7 ) and ( 3√5 , − 2√7 )
  • d = √{[ 3√5 − ( − 11√5 ) ]2 + ( − 2√7 − 4√7 )2}
  • d = √{( 14√5 )2 + ( − 6√7 )2}
  • d = √{980 + 252}
d = √{1232} = √{4 ×4 ×7 ×11} = 4√{77}
Find the distance between the points ( − 8√2 ,12√3 ) and ( − 5√2 ,9√3 )
  • d = √{[ ( − 5√2 ) − ( − 8√2 ) ]2 + ( 9√3 − 12√3 )2}
  • d = √{( 3√2 )2 + ( − 3√3 )2}
  • d = √{18 + 27}
d = √{45} = 3√5
Find x if the distance between the points ( 2√3 ,5√2 ) and ( x,7√2 ) is 8.
  • 8 = √{( x − 2√3 )2 + ( 7√2 − 5√2 )2}
  • 8 = √{( x − 2√3 )2 + ( 2√2 )2}
  • 8 = √{( x − 2√3 )2 + 8}
  • 84 = ( x − 2√3 )2 + 8
  • 56 = ( x − 2√3 )2
  • ±√{56} = x − 2√3
x = 2√3 ±√{56}

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Distance Formula

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Distance Formula 0:09
    • Similarity to Pythagorean Theorem
  • Missing Coordinates 5:50
    • Example
  • Example 1: Distance Between Points 11:43
  • Example 2: Distance Between Points 14:05
  • Example 3: Distance Between Points 18:18
  • Example 4: Missing Coordinate 21:57