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Understanding Distance and Midpoint in Coordinate Geometry

This lesson explores the concepts of distance and midpoint between two points in a Cartesian plane. We will start with the distance formula, derived from the Pythagorean theorem, illustrating how to calculate the distance between any two points (x1, y1) and (x2, y2). Additionally, we will define the midpoint as the average of the coordinates of the endpoints. The lesson includes practical examples and exercises, such as finding the distance between points (3, 6) and (8, 10), as well as determining midpoints and solving for unknown endpoints, enhancing your understanding of spatial relationships in geometry.

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Understanding Distance and Midpoint in Coordinate Geometry

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  1. Distance And Midpoint Section 1-3 Spi.2.1.E Jim Smith JCHS

  2. The distance between 2 points is the absolute value of the difference of the coordinates. The distance between exit 417 and 407 is | 417 – 407 | = 10 or | 407 – 417 | = | -10 | = 10

  3. A B | | | | | | | | | | | | | | -5 4 The distance between A and B is | -5 – 4 | = | -9 | = 9 Our distances should always be positive

  4. A B | | | | | | | | | | | | | | -6 12 The midpoint of a segment is the average of the coordinates 6 2 -6 + 12 2 = 3 =

  5. Review Graphing y Order ( X,Y ) Positive x ( 0,0 ) Origin Negative

  6. The Distance Formula Is Derived From The Pythagorean Formula A B

  7. Dist = ( x - x )² + ( y - y )² Distance Formula Remember the order ( x , y ) Check yourself … our answers should be positive

  8. ( 8 – 3 )² + ( 10 – 6 )² ( 5 )² + ( 4 )² 25 + 16 41 =6.40 Find the distance between: ( 3 , 6 ) and ( 8 , 10 ) ( 3 – 8 )² + ( 6 - 10 )² ( -5 )² + ( -4 )² 25 + 16 41 = 6.40

  9. MIDPOINT The midpoint of a segment is half way between the x’s and half way between the y’s You can call it the average Midpoint 6 10

  10. Midpoint Formula X + X , Y + Y 2 2 Find the midpoint of ( 2,8 ) and ( 6,4 ) 2 + 6 , 8 + 4 = 8 ,12 = ( 4 , 6 ) 2 2 2 2

  11. X1 + X2 2 =XMID Y1 + Y2 2 = YMID What If We Knew The Midpoint Of A Segment And One Endpoint? How Would We Find The Other Endpoint? Think Of The Formula As: Endpoints Midpoint (X1 , Y1)(X2 , Y2)( Xmid , Ymid)

  12. X1 + X2 2 Y1 + Y2 2 =XMID = YMID Endpoint ( 3 , 5 ) Midpoint ( 6 , -2 ) Find The Other Endpoint. Find ( X2 ,Y2 ) 5 + Y2 2 5 + Y2 Y2 = -9 3 + X2 2 3 + X2 X2 = 9 = -2 = 6 = -4 = 12 ( 9 , -9 )

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