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© Boardworks of 11 Distance in the Coordinate Plane
© Boardworks of 11 Information
© Boardworks of 11 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 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. The coordinates of a point P are written in the form P ( x, y ). The Cartesian coordinate system
© Boardworks of 11 a c b The area of the largest square is c × c or c 2. The areas of the smaller squares are a 2 and b 2. The Pythagorean theorem can be written as: c 2 = a 2 + b 2 a2a2 c2c2 b2b2 is equal to the sum of the squares of the lengths of the legs. The Pythagorean theorem: In a right triangle, the square of the length of the hypotenuse Reviewing the Pythagorean theorem
© Boardworks of 11 What is the distance between two general points with coordinates A ( x a, y a ) and B ( x b, y b )? horizontal distance between the points: vertical distance between the points: using the Pythagorean theorem, the square of the distance between the points A ( x a, y a ) and B ( x b, y b ) is: The distance formula: The distance formula ( x b – x a ) 2 + ( y b – y a ) 2 taking the square root gives AB : AB = ( x a – x b ) 2 + ( y a – y b ) 2 y b – y a x b – x a
© Boardworks of 11 write the distance formula: AB = ( x a – x b ) 2 + ( y a – y b ) 2 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). = = = 80 = 8.94 units (to the nearest hundredth) AB = (7–3) 2 + (6–(–2)) 2 substitute values:
© Boardworks of 11 The midpoint, M, of two points A ( x a, y a ) and B ( x b, y b ) is mean of each coordinate of the two points: is the mean of the x -coordinates. is the mean of the y -coordinates. Finding the midpoint x a + x b y a + y b 2 2
© Boardworks of 11 –3 + a 4 + b 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 ). = (1, –2) equating the x -coordinates: –3 + a = 2 a = 5 equating the y -coordinates: 4 + b = –4 b = –8 The coordinates of the point P are (5, –8). Using midpoint midpoint formula: –3 + a 4 + b 22 2 = 1 2 = –2
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