Co-ordinate Geometry

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Coordinate Geometry is a vital aspect of Further Mathematics that deals with the study of geometric figures using algebraic methods. It connects algebra and geometry, allowing us to represent geometric shapes and solve problems using coordinates and equations. The objectives of this topic are to help students understand the concept of coordinates in a plane, calculate the midpoint of a line segment, determine coordinates that divide a line in a given ratio, calculate distances between points, find gradients of lines, and derive equations for lines.

Coordinates in a Plane: To begin with, understanding coordinates in a plane is fundamental to coordinate geometry. In a two-dimensional plane, a point is uniquely identified by its coordinates – an ordered pair (x, y). The x-coordinate represents the horizontal position, whereas the y-coordinate shows the vertical position. These coordinates are essential for plotting points, defining shapes, and solving geometric problems.

Midpoint of a Line Segment: One of the key concepts in coordinate geometry is determining the midpoint of a line segment. The midpoint M of a line segment AB is the point that divides the segment into two equal parts. To find the midpoint, we take the average of the x-coordinates and the y-coordinates of the endpoints A and B. This midpoint formula helps us locate the center point of a line segment.

Dividing a Line in a Given Ratio: Apart from finding midpoints, coordinate geometry enables us to locate points that divide a line segment in a given ratio. Given two points A(x₁, y₁) and B(x₂, y₂), we can calculate the coordinates of a point P that divides AB in the ratio m:n. By applying the section formula, we can find the precise coordinates of the dividing point.

Distance Between Two Points: In coordinate geometry, measuring the distance between two points A and B is crucial for determining lengths, perimeters, and other geometric properties. The distance formula, derived from the Pythagorean theorem, allows us to calculate the distance AB using the coordinates of the two points. This formula is applicable across various geometric contexts.

Gradient of a Line: Another significant aspect of coordinate geometry is the concept of gradient, which represents the slope of a line. The gradient is calculated as the ratio of the change in y-coordinates to the change in x-coordinates between two points on the line. It indicates the steepness of the line and is crucial for understanding the direction and inclination of lines.

Equation of a Line: Finally, deriving the equation of a line from its gradient and a point on the line is a key skill in coordinate geometry. The point-slope form or slope-intercept form can be used to find the equation of a line when the gradient and a point are given. This equation serves as a mathematical representation of the line and allows for further analysis and problem-solving.

Objetivos

  1. Calculate the distance between two points on a plane
  2. Find the gradient of a line given two points on the line
  3. Derive the equation of a line given the gradient and a point on the line
  4. Calculate the midpoint of a line segment
  5. Understand the concept of coordinates in a plane
  6. Determine the coordinates of points that divide a line in a given ratio

Nota de Aula

Co-ordinate Geometry, also known as Cartesian Geometry, is a branch of mathematics that uses algebraic equations to describe the positions of points, lines, and shapes in a plane. It combines algebra and geometry to solve real-world problems. The basic element of co-ordinate geometry is the point, which is defined by an ordered pair of numbers (x, y). These numbers are called the coordinates of the point.

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  1. Calculate the midpoint of the line segment with endpoints at (3,5) and (-1,7). A. (1,6) B. (2,-1) C. (0,6) D. (7,3) Answer: A. (1,6)
  2. Given a line segment with endpoints A(2,1) and B(8,5), determine the coordinates of a point C that divides AB in the ratio 2:1. A. (3,2) B. (6,4) C. (5,3) D. (4,3) Answer: B. (6,4)
  3. Find the distance between the points P(4,3) and Q(9,7). A. √17 B. 7 C. 5 D. √29 Answer: A. √17
  4. If the gradient of a line is 2 and it passes through the point (1,3), what is the equation of the line in the form y = mx + c? A. y = 2x + 1 B. y = 2x - 1 C. y = 3x + 1 D. y = 3x - 2 Answer: A. y = 2x + 1
  5. Determine the equation of a line that passes through the points (2,3) and (4,1). A. y = -2x + 7 B. y = 2x - 1 C. y = -2x + 5 D. y = 2x - 5 Answer: C. y = -2x + 5
  6. Calculate the coordinates of the point that divides the line segment joining (1,5) and (3,9) internally in the ratio 3:2. A. (2,7) B. (3,8) C. (5,6) D. (4,7) Answer: A. (2,7)
  7. Find the gradient of the line passing through the points (5,4) and (7,9). A. 1.5 B. 2 C. 2.5 D. 3 Answer: A. 1.5
  8. Determine the equation of a line parallel to y = 3x + 1 passing through the point (2,5). A. y = 3x - 1 B. y = 3x + 2 C. y = -3x + 2 D. y = -3x + 1 Answer: B. y = 3x + 2
  9. If the midpoint of a line segment with endpoints at (-3,7) and (5,-1) is (-1,3), what are the coordinates of the other endpoint? A. (7,-7) B. (9,-11) C. (3,3) D. (-5,7) Answer: A. (7,-7)

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Perguntas Anteriores

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WAEC SSCE - Further Mathematics - 2022 (Clique para fazer a avaliação completa.)

Pergunta 1 Relatório

Solve 3cos2x - sinx = 0 for 0º≤x≤360º

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Pratica uma série de Co-ordinate Geometry perguntas anteriores