{"id":143227,"date":"2022-07-16T23:23:01","date_gmt":"2022-07-16T17:53:01","guid":{"rendered":"https:\/\/ccssanswers.com\/?p=143227"},"modified":"2022-06-30T14:37:56","modified_gmt":"2022-06-30T09:07:56","slug":"worksheet-on-collinearity-of-three-points","status":"publish","type":"post","link":"https:\/\/ccssanswers.com\/worksheet-on-collinearity-of-three-points\/","title":{"rendered":"Worksheet on Collinearity of Three Points | Collinearity of Three Points Worksheet PDF"},"content":{"rendered":"<p>Worksheet on Collinearity of Three Points with step-by-step solutions is available on this page. Thus the students who are interested to learn the concept of <a href=\"https:\/\/ccssanswers.com\/conditions-of-collinearity-of-three-points\/\">conditions of collinearity of three points<\/a> and solving the problems can refer to this article. By practicing the problems from the given Collinearity of Three Points worksheet you can know different types of questions asked in the exams and know how to solve them.<\/p>\n<p>Also Read:<\/p>\n<ul>\n<li><a href=\"https:\/\/ccssanswers.com\/distance-and-section-formulae\/\">Distance and Section Formulae<\/a><\/li>\n<li><a href=\"https:\/\/ccssanswers.com\/worksheet-on-distance-formula\/\">Worksheet on Distance Formula<\/a><\/li>\n<li><a href=\"https:\/\/ccssanswers.com\/distance-formula-in-geometry\/\">Distance Formula in Geometry<\/a><\/li>\n<\/ul>\n<h2>Collinearity of Three Points Worksheet with Answers<\/h2>\n<p><strong>Example 1.<\/strong><br \/>\nFind the value of p for which the points (p, -1), (4, 1), and (3, 5) are collinear.<\/p>\n<details>\n<summary><strong>Solution:<\/strong><\/summary>\n<p>Let the given points be:<br \/>\nA(p, -1) = (x<sub>1<\/sub>, y<sub>1<\/sub>)<br \/>\nB(4, 1) = (x<sub>2<\/sub>, y<sub>2<\/sub>)<br \/>\nC(3, 5) = (x<sub>3<\/sub>, y<sub>3<\/sub>)<br \/>\nGiven that A, B, and C are collinear.<br \/>\nSlope of AB = Slope of BC<br \/>\n(y<sub>2<\/sub> &#8211; y<sub>1<\/sub>)\/(x<sub>2<\/sub> &#8211; x<sub>1<\/sub>) = (y<sub>3<\/sub> &#8211; y<sub>2<\/sub>)\/(x<sub>3<\/sub> &#8211; x<sub>2<\/sub>)<br \/>\nSubstituting the values of coordinates of given points,<br \/>\n(1 + 1)\/(4 &#8211; p) = (5 &#8211; 1)\/(3 &#8211; 2)<br \/>\n2\/(4 &#8211; p) = 4\/1<br \/>\n2\/(4 &#8211; p) = 4<br \/>\n4 &#8211; p = 8<br \/>\np = 4 &#8211; 8<br \/>\np = -4<br \/>\nHence, the value of p is -4.<\/p>\n<\/details>\n<hr \/>\n<p><strong>Example 2.<\/strong><br \/>\nUsing the equation method, check the collinearity of the points A(6,-4), B(1,1), and C(-1,6).<\/p>\n<details>\n<summary><strong>Solution:<\/strong><\/summary>\n<p>We know that the equation of a line passing through the points (x<sub>1<\/sub>, y<sub>1<\/sub>) and (x<sub>2<\/sub>, y<sub>2<\/sub>) is:<br \/>\ny \u2013 y<sub>1<\/sub> = [(y<sub>2<\/sub> \u2013 y<sub>1<\/sub>) \/(x<sub>2<\/sub> \u2013 x<sub>1<\/sub>)] (x \u2013 x<sub>1<\/sub>)<br \/>\nLet A(6, -4) = (x<sub>1<\/sub>, y<sub>1<\/sub>) and B(1,1) = (x<sub>2<\/sub>, y<sub>2<\/sub>).<br \/>\nSo, the equation of a line passing through the points A(6, -4) and B(1, 1) is given by:<br \/>\ny + 2 = [(1 + 4)\/(1 \u2013 6)] (x \u2013 7)<br \/>\ny + 2 = (5\/-5) (x \u2013 7)<br \/>\ny + 2 = -x + 7<br \/>\nx + y + 2 \u2013 7 = 0<br \/>\nx + y \u2013 5 = 0<br \/>\nNow, substituting the point C(-1, 6) in the above equation,<br \/>\n-1 + 6 \u2013 5 = 0<br \/>\n0 = 0<br \/>\nThus, the third point satisfies the equation of the line passing through the two of the given three points.<br \/>\nTherefore, the given points A, B and C are collinear.<\/p>\n<\/details>\n<hr \/>\n<p><strong>Example 3.<\/strong><br \/>\nProve that the three points A(2,2), B(6,4), and C(10,6) are collinear.<\/p>\n<details>\n<summary><strong>Solution:<\/strong><\/summary>\n<p>Given three points A(2,2),B(6,4), and C(10,6) are collinear, then the slopes of any two pairs of points will be equal.<br \/>\nThat is,<br \/>\nSlope of AB = 6 &#8211; 2\/4 &#8211; 2 = 4\/2 = 2<br \/>\nSlope of BC=10 &#8211; 6\/6 &#8211; 4 = 4\/2 = 2<br \/>\nSlope of AC=10 -2\/6 &#8211; 2 = 8\/4 = 2<br \/>\nThus, slope of AB = Slope of BC = Slope of AC<br \/>\nHence, proved.<\/p>\n<\/details>\n<hr \/>\n<p><strong>Example 4.<\/strong><br \/>\nThe points A(10,1),B(15,7) and C(a,3) are collinear. Find a.<\/p>\n<details>\n<summary><strong>Solution:<\/strong><\/summary>\n<p>The given points A,B and C are collinear<br \/>\nWe know that the slope of the line segment is slope = m = y2\u2212y1\/x2\u2212x1<br \/>\nThe slope of AB= slope of BC<br \/>\n7\u22121\/15\u221210 = 3\u22127\/a\u221215<br \/>\n6\/5 = -4\/a\u221215<br \/>\n6(a\u221215) = -4\u00d75<br \/>\n6a\u221290 = -20<br \/>\n6a = -20+90<br \/>\n6a = 70<br \/>\na=70\/6<br \/>\n=70\/6<br \/>\nHence, the value of a is 70\/6.<\/p>\n<\/details>\n<hr \/>\n<p><strong>Example 5.<\/strong><br \/>\nUsing the equation method, check the collinearity of the points A(1,-4), B(1,2) and C(-4,6).<\/p>\n<details>\n<summary><strong>Solution:<\/strong><\/summary>\n<p>We know that the equation of a line passing through the points (x1, y1) and (x2, y2) is:<br \/>\ny \u2013 y<sub>1<\/sub> = [(y<sub>2<\/sub> \u2013 y<sub>1<\/sub>) \/(x<sub>2<\/sub> \u2013 x<sub>1<\/sub>)] (x \u2013 x<sub>1<\/sub>)<br \/>\nLet A(1, -4) = (x<sub>1<\/sub>, y<sub>1<\/sub>) and B(1,2) = (x<sub>2<\/sub>,y<sub>2<\/sub>).<br \/>\nSo, the equation of a line passing through the points A(1, -4) and B(1, 2) is given by:<br \/>\ny + 2 = [(2 + 4)\/(1 \u2013 1)] (x \u2013 7)<br \/>\ny + 2 = (6\/0) (x \u2013 7)<br \/>\ny + 2 = 6x + 42<br \/>\n-6x + y + 2 &#8211; 42 = 0<br \/>\n-6x + y &#8211; 40 = 0<br \/>\nNow, substituting the point C(-4, 6) in the above equation,<br \/>\n-6(-4) + 6 &#8211; 5 = 0<br \/>\n-24+1 = 0<br \/>\n-23 = 0<br \/>\nThus, the third point does not satisfy the equation of the line passing through two of the given three points.<br \/>\nTherefore, the given points A, B, and C are not collinear.<\/p>\n<\/details>\n<hr \/>\n<p><strong>Example 6.<\/strong><br \/>\nProve that the three points A(4,2), B(5,4), and C(7,6) are collinear.<\/p>\n<details>\n<summary><strong>Solution:<\/strong><\/summary>\n<p>Given three points A(4,2),B(5,4), and C(7,6) are collinear, then the slopes of any two pairs of points will be equal.<br \/>\nThat is,<br \/>\nSlope formula = y<sub>2<\/sub> &#8211; y<sub>1<\/sub>\/x<sub>2<\/sub> &#8211; x<sub>1<\/sub><br \/>\nSlope of AB = 4-2\/ 5-4 = 2<br \/>\nSlope of BC= 6-4\/ 7-5 7 = 1<br \/>\nSlope of AC= 6-2\/ 7-4 = 4\/3<br \/>\nThus, slope of AB \u2260 Slope of BC \u2260 Slope of AC<br \/>\nHence, the given points are not collinear.<\/p>\n<\/details>\n<hr \/>\n<p><strong>Example 7.<\/strong><br \/>\nThe points A(2,3), B(4,5) and C(a,6) are collinear. Find a.<\/p>\n<details>\n<summary><strong>Solution:<\/strong><\/summary>\n<p>The given points A,B and C are collinear<br \/>\nWe know that the slope of the line segment is slope = m = y<sub>2<\/sub> &#8211; y<sub>1<\/sub>\/x<sub>2<\/sub>\u2212x<sub>1<\/sub><br \/>\nThe slope of AB= slope of BC<br \/>\n5-3\/ 4-2= 6 &#8211; 5\/a &#8211; 4<br \/>\n2\/2 = 1\/a-4<br \/>\n1 = 1\/a &#8211; 4<br \/>\na &#8211; 4 = 1<br \/>\na = 4 &#8211; 1<br \/>\na = -3<br \/>\nHence, the value of a is -3.<\/p>\n<\/details>\n<hr \/>\n<p><strong>Example 8.<\/strong><br \/>\nShow that the points (1,-1), (5,2) and (9,5) are collinear<\/p>\n<details>\n<summary><strong>Solution:<\/strong><\/summary>\n<p>Given that the points are<br \/>\nA(1,-1)<br \/>\nB(5,2)<br \/>\nC(9,5)<br \/>\nWe know that the distance formula is\u221a(x<sub>2<\/sub>\u2212x<sub>1<\/sub>)\u00b2+(y<sub>2<\/sub> &#8211; y<sub>1<\/sub>)\u00b2<br \/>\nDistance of AB = \u221a(5-1)\u00b2 + (2+1)\u00b2 = \u221a16 + 9 = \u221a25 = 5<br \/>\nDistance of BC = \u221a(5-9)\u00b2 + (2-5)\u00b2 = \u221a16+9 = \u221a25 = 5<br \/>\nDistance of AC = \u221a(1-9)\u00b2 + (-1-5)\u00b2 = \u221a64 + 36 = \u221a100 = 10<br \/>\nFrom these distance od AB+ distance of BC = distance of AC<br \/>\nSo the given points are collinear.<\/p>\n<\/details>\n<hr \/>\n<p><strong>Example 9.<\/strong><br \/>\nShow that the given points (1,2), (3,2), and (6,2) are collinear or not.<\/p>\n<details>\n<summary><strong>Solution:<\/strong><\/summary>\n<p>Given that the points are<br \/>\nA(1,-1)<br \/>\nB(5,2)<br \/>\nC(9,5)<br \/>\nWe know that the distance formula is \u221a(x<sub>2<\/sub>\u2212x<sub>1<\/sub>)\u00b2+(y<sub>2<\/sub> &#8211; y<sub>1<\/sub>)\u00b2<br \/>\nDistance of AB = \u221a(3-1)\u00b2 + (2-2)\u00b2 = \u221a4 = 2<br \/>\nDistance of BC = \u221a(6-3)\u00b2 + (2-2)\u00b2 = \u221a9 = 3<br \/>\nDistance of AC = \u221a(6-1)\u00b2 + (2-2)\u00b2 = \u221a25 = 5<br \/>\nFrom these distance od AB+ distance of BC = distance of AC<br \/>\nSo the given points are collinear.<\/p>\n<\/details>\n<hr \/>\n<p><strong>Example 10.<\/strong><br \/>\nFind the value of p for which the points (k, -2), (3, 4), and (4, 5) are collinear.<\/p>\n<details>\n<summary><strong>Solution:<\/strong><\/summary>\n<p>Let the given points be:<br \/>\nA(k, -2) = (x<sub>1<\/sub>, y<sub>1<\/sub>)<br \/>\nB(3, 4) = (x<sub>2<\/sub>, y<sub>2<\/sub>)<br \/>\nC(4, 5) = (x<sub>3<\/sub>, y<sub>3<\/sub>)<br \/>\nGiven that A, B, and C are collinear.<br \/>\nSlope of AB = Slope of BC<br \/>\n(y<sub>2<\/sub> &#8211; y<sub>1<\/sub>)\/(x<sub>2<\/sub>\u2212x<sub>1<\/sub>) = (y<sub>3<\/sub> &#8211; y<sub>2<\/sub>)\/(x<sub>3<\/sub> &#8211; x<sub>2<\/sub>)<br \/>\nSubstituting the values of coordinates of given points,<br \/>\n(4 + 2)\/(3 &#8211; p) = (5 &#8211; 3)\/(4 &#8211; 4)<br \/>\n6\/(3 &#8211; p) = 2<br \/>\n6\/(3 &#8211; p) = 2<br \/>\n3 &#8211; p = 12<br \/>\n-p = 12 &#8211; 3<br \/>\np = -9<br \/>\nHence, the value of p is -9.<\/p>\n<\/details>\n<hr \/>\n","protected":false},"excerpt":{"rendered":"<p>Worksheet on Collinearity of Three Points with step-by-step solutions is available on this page. Thus the students who are interested to learn the concept of conditions of collinearity of three points and solving the problems can refer to this article. By practicing the problems from the given Collinearity of Three Points worksheet you can know &#8230; <a title=\"Worksheet on Collinearity of Three Points | Collinearity of Three Points Worksheet PDF\" class=\"read-more\" href=\"https:\/\/ccssanswers.com\/worksheet-on-collinearity-of-three-points\/\" aria-label=\"More on Worksheet on Collinearity of Three Points | Collinearity of Three Points Worksheet PDF\">Read more<\/a><\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5],"tags":[],"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/posts\/143227"}],"collection":[{"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/comments?post=143227"}],"version-history":[{"count":1,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/posts\/143227\/revisions"}],"predecessor-version":[{"id":143228,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/posts\/143227\/revisions\/143228"}],"wp:attachment":[{"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/media?parent=143227"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/categories?post=143227"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/tags?post=143227"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}