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Distance in the Coordinate Plane

Distance in the Coordinate Plane. Information. The Cartesian coordinate system. The Cartesian coordinate system is named after the French mathematician Ren é Descartes (1596 – 1650).

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Distance in the Coordinate Plane

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  1. Distance in the Coordinate Plane

  2. Information

  3. The Cartesian coordinate system The Cartesian coordinate system is named after the French mathematician René Descartes (1596 – 1650). Points in the (x, y) plane are defined by their perpendicular distance from the x- and y-axes relative to the origin, O. The coordinates of a point P are written in the form P(x, y). The x-coordinate gives the horizontal distance from the y-axis to the point. The y-coordinate gives the vertical distance from the x-axis to the point.

  4. Reviewing the Pythagorean theorem The Pythagorean theorem: In a right triangle, the square of the length of the hypotenuse is equal tothe sum of the squares of the lengths of the legs. The area of the largest square is c × c or c2. c2 The areas of the smaller squares are a2 and b2. c a2 a The Pythagorean theorem can be written as: b b2 c2 =a2 + b2

  5. The distance between two points Given the coordinates of two points, A and B, the distance between them is found by the Pythagorean theorem.

  6. The distance formula What is the distance between two general points with coordinates A(xa, ya) and B(xb, yb)? horizontal distance between the points: xb – xa vertical distance between the points: yb – ya using the Pythagorean theorem, the square of the distance between the points A(xa, ya) and B(xb, yb) is: (xb – xa)2 + (yb – ya)2 taking the square root gives AB: AB = √(xa – xb)2 + (ya – yb)2 The distance formula:

  7. Using the distance formula The distance formula is used to find the distance between any two points on a coordinate grid. Find the distance between the points A(7,6) and B(3,–2). write the distance formula: AB = √(xa – xb)2 + (ya – yb)2 substitute values: AB = √(7–3)2 + (6–(–2))2 = 42 + 82 = 16 + 64 = √80 = 8.94 units (to the nearest hundredth)

  8. Find the lengths

  9. Finding the midpoint The midpoint, M, of two points A(xa, ya) and B(xb, yb) is mean of each coordinate of the two points: xa + xb is the mean of the x-coordinates. 2 ya + yb is the mean of the y-coordinates. 2

  10. The midpoint of a line segment

  11. Using midpoint The midpoint of the line segment joining the point (–3, 4) to the point P is (1, –2). Find the coordinates of the point P. Let the coordinates of the points P be (a, b). –3 + a 4 + b = (1, –2) midpoint formula: 2 2 equating the x-coordinates: equating the y-coordinates: –3 + a 4 + b = 1 = –2 2 2 –3 + a = 2 4 + b = –4 a = 5 b = –8 The coordinates of the point P are (5, –8).

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