{"id":4078,"date":"2023-10-06T09:27:29","date_gmt":"2023-10-06T09:27:29","guid":{"rendered":"http:\/\/localhost\/ecole9ja\/?p=4078"},"modified":"2023-10-06T09:30:04","modified_gmt":"2023-10-06T09:30:04","slug":"week-3-ss3-second-term-mathematics-notes","status":"publish","type":"post","link":"https:\/\/ecolebooks.com\/nigeria\/posts\/week-3-ss3-second-term-mathematics-notes\/","title":{"rendered":"Week 3  &#8211; SS3 Second Term Mathematics Notes"},"content":{"rendered":"<p>\u00a0<strong>WEEK THREE<br \/>\n\t\t\t<\/strong><\/p>\n<ul>\n<li>Coordinate Geometry of straight lines:\n<\/li>\n<li>Gradient and Intercepts of a line\n<\/li>\n<li>Angle between two intersecting straight lines and application\n<\/li>\n<\/ul>\n<p>\u00a0<strong>Gradient and Intercepts of a line<\/strong><br \/>\n\tGradient of a line of the form y = mx + c, is the coefficient of x, which is represented by m and c is the intercept on the y axis.<br \/>\n<strong>Example<br \/>\n<\/strong>1. Find the equation of the line with gradient 4 and y-intercept -7.<br \/>\n<strong>Solution<br \/>\n<\/strong>m = 4, c = &#8211; 7,<br \/>\nHence, the equation is; y =4x &#8211; 7.<\/p>\n<p>\u00a0<strong>Evaluation:<br \/>\n<\/strong>1.What is the gradient and y intercept of the line equation 3x -5y +10=0 ?<br \/>\n2. Find the equation of the line with gradient &#8211; 9 and y-intercept 4.<\/p>\n<p>\u00a0<strong>Gradient and One Point Form<br \/>\n<\/strong>The equation of the line can be calculated given one point (x, y) and gradient (m) by using the formula; y &#8211; y1= m(x &#8211; x1)<\/p>\n<p>\u00a0<strong>Example<br \/>\n<\/strong>Find the equation of the line with gradient -8 and point(3, 7).<br \/>\n<strong>Solution<br \/>\n<\/strong>m = &#8211; 8, (x1, y1) =(3,7)<br \/>\nEquation: y &#8211; 7 = &#8211; 8(x &#8211; 3)<br \/>\n                 y = -8x + 24 +7<br \/>\n                 y = -8x + 31<\/p>\n<p>\u00a0<strong>Evaluation:<br \/>\n<\/strong>1. Find the equation of the line with gradient 5 and point(-2, -7).<br \/>\n2. Find the equation of the line with gradient -12and point (3, -5).<\/p>\n<p>\u00a0<strong>Two Point Form:<br \/>\n<\/strong>Given two points (x1, y1) and (x2, y2), the equation can be obtained using the formula:<br \/>\n y2 &#8211; y1 = y &#8211; y1<br \/>\nx2 &#8211; x1      x &#8211; x1<br \/>\nExample: Find the equation of the line passing through (2,-5) and (3,6).<br \/>\n<strong>Solution<br \/>\n<\/strong>6 &#8211; (-5)\/3 &#8211; 2 = y &#8211; (-5)\/x &#8211; 2<br \/>\n11 = y + 5\/x &#8211; 2<br \/>\n11(x &#8211; 2) = y + 5<br \/>\n11x &#8211; 22 = y + 5<br \/>\ny &#8211; 11x + 27 = 0<\/p>\n<p>\u00a0<strong>Evaluation:<br \/>\n<\/strong>1.Find the equation of the line passing through (3, 4) and (-1, -2).<br \/>\n2.Find the equation of the line passing through (-8, 5) and (-6, 2).<\/p>\n<p>\u00a0<strong>Angles between Lines<br \/>\n<\/strong><strong>Parallel lines:<br \/>\n<\/strong>The angle between parallel lines is 0<sup>0<\/sup> because they have the same gradient<\/p>\n<p>\u00a0<strong>Perpendicular Lines:<br \/>\n<\/strong>Angle between two perpendicular lines is 90<sup>0<\/sup> and the product of their gradients is \u2013 1. Hence, m<sub>1<\/sub>m<sub>2<\/sub> = &#8211; 1<br \/>\n<strong>Examples:<br \/>\n<\/strong><\/p>\n<ol>\n<li>Show that the lines y = -3x + 2 and y + 3x = 7 are parallel.\n<\/li>\n<\/ol>\n<p>solution:<br \/>\n         Equation 1: y = -3x + 2,   m<sub>1<\/sub> = -3<br \/>\n         Equation 2:  y + 3x = 7,<br \/>\n                                 y = -3x + 7, m<sub>2<\/sub> = &#8211; 3<br \/>\nsince; m<sub>1<\/sub> = m<sub>2<\/sub> = &#8211; 3, then the lines are parallel<\/p>\n<ol>\n<li>Given the line equations x = 3y + 5 and y + 3x = 2, show that the lines are perpendicular.\n<\/li>\n<\/ol>\n<p>solutions:<br \/>\n     Equation 1:     x = 3y + 5,   make y the subject of the equation.<br \/>\n                              3y = x + 5<br \/>\n                                y = x\/3 + 5\/3<br \/>\n                            m<sub>1<\/sub> = 1\/3<br \/>\n  Equation 2:    y + 3x = 2,<br \/>\n                              y = &#8211; 3x + 2,   m<sub>2<\/sub> = -3<br \/>\nhence: m<sub>1<\/sub> x m<sub>2<\/sub> = 1\/3 x \u2013 3 = &#8211; 1<br \/>\nsince: m<sub>1<\/sub>m<sub>2<\/sub> = &#8211; 1, then the lines are perpendicular.<\/p>\n<p>\u00a0<strong>Evaluation: <\/strong>State which of the following pairs of lines are: (i) perpendicular   (ii) parallel<br \/>\n             (1)    y = x + 5 and y = &#8211; x + 5      (2). 2y \u2013 6 = 5x and 3 \u2013 5y = 2x    (3) y = 2x \u2013 1 and 2y \u2013 4x = 8 <\/p>\n<p>\u00a0<strong>Angles between Intersecting Lines:<br \/>\n<\/strong><img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100623_0927_Week3and41.png\" alt=\"\"\/><strong>y<br \/>\n<\/strong><img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100623_0927_Week3and42.png\" alt=\"\"\/><br \/>\n\t\t                                  y = mx + c               <\/p>\n<p>\u00a0<br \/>\n\u00a0<img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100623_0927_Week3and43.png\" alt=\"\"\/>                    \u03b8                                                x<br \/>\n                        0   <\/p>\n<p>\u00a0The gradient of y = mx + c is tan \u03b8.    Hence <strong>m = tan \u03b8<\/strong>can be used to calculate angles between two intersecting lines. Generally the angle between two lines can be obtained using: tan 0 = m2 -m1<br \/>\n                                                                                                                                                                 1 + m1m2<br \/>\nExample: Calculate the acute angle between the lines y=4x -7 and y = x\/2 + 0.5.<br \/>\nSolution:<br \/>\nY=4x -7, m1= 4, y=x\/2+0.2, m2 =1\/2.<br \/>\n Tan O= 0.5 &#8211; 4.       = -3.5\/3<br \/>\n                    1 + (0.5*4)<br \/>\nTan O =- 1.1667<br \/>\nO=tan-1(-1.1667) = 49.4<\/p>\n<p>\u00a0Evaluation:Calculate the acute angle between the lines y=3x -4 and x &#8211; 4y +8 = 0.<\/p>\n<p>\u00a0General Evaluation:<br \/>\n1.Calculate the acute angle between the lines y=2x -1 and  2y + x = 2.<br \/>\n2.If the lines 3y=4x -1 and qy= x + 3 are parallel to each other, find the value of q.<br \/>\n3.Find the equation of the line passing through (2,-1) and gradient 3.<\/p>\n<p>\u00a0<strong>Reading Assignment: NGM for SS 3 Chapter 9 <\/strong>page 75 \u2013 81 <\/p>\n<p>\u00a0Weekend Assignment<br \/>\n1.Find the equation of the line passing through (5,0) and gradient 3.<br \/>\n2.Find the equation of the line passing through (2,-1) and (1, -2).<br \/>\n 3. Two lines y=3x &#8211; 4 and x &#8211; 4y + 8=0 are drawn on the same axes.<br \/>\n(a) Find the gradients and intercepts on the axes of each line.<br \/>\n(b) Find the equation parralel to x -4y + 8=0 at the point (3, -5)<br \/>\n\u00a0\u00a0\u00a0\u00a0<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u00a0WEEK THREE Coordinate Geometry of straight lines: Gradient and Intercepts of a line Angle between&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1,316],"tags":[],"class_list":["post-4078","post","type-post","status-publish","format-standard","hentry","category-posts","category-second-term-ss3-mathematics"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/4078","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/comments?post=4078"}],"version-history":[{"count":2,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/4078\/revisions"}],"predecessor-version":[{"id":4082,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/4078\/revisions\/4082"}],"wp:attachment":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/media?parent=4078"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/categories?post=4078"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/tags?post=4078"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}