Analytic Geometry الھندسة التحليلية نظام اإلحداثيات الديكارتي 1-1 Cartesian Coordinate System The Cartesian coordinate system, or the rectangular coordinate system, is a geometrical system that is used to determine the locations of points in a plane. Points are located with respect to a reference point called the origin which is the intersection point of a horizontal line, known as x-axis, and a vertical line called y-axis. The x and y axes divide the Cartesian plane into four regions called quadrants. Each point in the plane is defined by an ordered pair (x, y) of real numbers called the coordinates of the point. An example of ordered pairs or coordinates is the point P below: 2
3 قانون إيجاد المسافة 1-2 The Distance Formula The distance d between two points A(x 1, y 1 ) and B(x 2, y 2 ) can be found from the distance formula: 4
Example 1: Find the distance between the points C(3, 4) and D( 13, 11). 5 قانون إيجاد نقطة المنتصف 1-3 The Midpoint Formula The coordinates of the midpoint of a line segment joining the two points A(x 1, y 1 ) and B(x 2, y 2 ) are found by averaging the coordinates of the endpoints. The midpoint formula is: 6
Example 2: F is the midpoint between points C(3, 4) and D( 13, 11). Find its coordinates. 7 ميل الخط المستقيم 1-4 The Slope of a Line The slope of a line is a measurement of its steepness and direction. Slope of a line m is calculated from the following formula which is called the slope formula: 8
Depending on the direction of the line, its slope could be positive, negative, zero or undefined and as shown below. 9 Example 3: Find the slope of the line that passes through points P( 4, 8) and R(9, 7). 10
Example 4: Find the slope of the lines a, b, c and d shown in the figure below. 11 الخطوط المتوازية 1-5 Parallel Lines 12
الخطوط المتعامدة 1-6 Perpendicular Lines 13 Example 5: Lines m and n are parallel. If the slope of line m is 0.48, what is the slope of line n? Since the two lines are parallel, then they have the same slope. So, the slope of line n = 0.48 Example 6: Line c is perpendicular to line d and the slope of line c is 0.5. Find the slope of line d. Since lines c and d are perpendicular, then their slopes are opposite reciprocals of one another. Therefore, the slope of line d = 14
المقطع السيني و المقطع الصادي 1-7 x-intercept and y-intercept For a non-horizontal line, x-intercept is the x-coordinate of the point where the line intersects x-axis. In the same way, for a non-vertical line, y-intercept is the y- coordinate of the point where it intersects y-axis. a 15 معادلة الخط المستقيم 1-8 Equations of Lines The equation of a line is a mathematical sentence that describes the relationship between the x-coordinate and the y-coordinate of all its points. The equation of a line is of the first degree and is therefore called a linear equation. Straight line equation may be written in any of the following three forms: where m is the slope. where m is the slope, and b is the y-intercept. 16
Example 7: Draw the graph of the line whose equation is 2x 3y = 6 using two randomly selected points. Let x = 1, then: Let y = 1, then: Thus, (1, 1.33) is the first point. Thus, (4.5, 1) is the second point. 17 The graph of 2x 3y = 6 is as shown in the figure below. 18
Example 8: Draw the graph of the line whose equation is 2x 3y = 6 using the x-intercept and the y-intercept. x-intercept y = 0 So, (3, 0) is the first point. y-intercept x = 0 Therefore, (0, 2) is the first point. 19 Example 9: Draw the graph of : (a) x = 4 (b) y = 2 (a) The graph of x = 4 is a vertical line with x-coordinate = 4 for all its points (b) The graph of y = 2 is a horizontal line with y-coordinate = 2 for all its points 20
Example 10: Find the equation of the line that passes though the points (4, 5) and ( 11, 3). Write the equation in point-slope form, standard form and slope-intercept form. Find the slope first: Write the equation in point-slope form: 21 To write the equation in the standard form, multiply both sides of the equation by 15: Rearrange the equation 8x + 15y = 43 to write it in the slope-intercept form: Divide both sides by 15: 22
Example 11: Determine whether the lines 6x + 4y = 9 and 8x 12y = 7 are parallel or perpendicular or neither. Find the slopes of the two lines and compare them: m1 = m2 = So, the two lines are perpendicular since their slopes are opposite reciprocals of one another. 23 Example 12: Which of the points A(2, 1.6) and B(1, 2.2) lie on the graph of the line 3x + 5y = 14? If a point lies on the graph of a line then it satisfies its equation. Check point A(2, 1.6): Check point B(1, 2.2) : Point A(2, 1.6) lies on the graph of 3x + 5y = 14 Point B(1, 2.2) doesn t lie on the graph of 3x + 5y = 14 24
معادلة الدائرة 1-9 Equations of Circles The radius r of a circle with a centre at the point (h, k) can be found using the distance formula between the centre and any point on the circle (x, y) and as follows: The standard form of the equation of a circle of radius r with centre at the point (h, k) is: (x h) 2 + (y k) 2 = r 2 25 Example 13: The point (3, 4) lies on a circle whose centre is at ( 1, 2), as shown in the Figure below. Write the standard form of the equation of this circle. The radius r of the circle is the distance between ( 1, 2) and (3, 4): Using (h, k) = ( 1, 2) and r, The equation of the circle is: Standard Form 26
تناظر المعادالت 1-10 Symmetry of Equations A graph has symmetry with respect to the y-axis if whenever (x, y) is on the graph, so is the point ( x, y). A graph has symmetry with respect to the origin if whenever (x, y) is on the graph, so is the point ( x, y). A graph has symmetry with respect to the x-axis if whenever (x, y) is on the graph, so is the point (x, y). 27 Example 14: Test y = x 2 + 2 for symmetry with respect to the x-axis, the y-axis, and the origin.. x-axis We replace y with y: y-axis We replace x with x: Multiplying both sides by 1: Simplifying gives: The resulting equation is not equivalent to the original equation, so the graph is not symmetric with respect to the x-axis. The resulting equation is equivalent to the original equation, so the graph is symmetric with respect to the y-axis. 28
Origin We replace x with x and y with y: Simplifying gives: The resulting equation is not equivalent to the original equation, so the graph is not symmetric with respect to the origin. 29 Example 15: Test x 2 + y 4 = 5 for symmetry with respect to the x-axis, the y-axis, and the origin.. x-axis We replace y with y: y-axis We replace x with x: The resulting equation is equivalent to the original equation, so the graph is symmetric with respect to the x-axis. The resulting equation is equivalent to the original equation, so the graph is symmetric with respect to the y-axis. 30
Origin We replace x with x and y with y: The resulting equation is equivalent to the original equation, so the graph is symmetric with respect to the origin. 31