An illustration showing the construction used to divide a line AB into two equal parts; and to erect a perpendicular through the middle. “With the end A and B as centers, draw the dotted circle arcs with a radius greater than half the line. Through the crossings of the arcs draw the perpendicular CD, which divides the line into two equal parts.” Large GIF 1024×913, 22.6 KiB Medium GIF 640×570, 12.6 KiB Small GIF 320×285, 5.7 KiB {"appState":{"pageLoadApiCallsStatus":true},"articleState":{"article":{"headers":{"creationTime":"2016-03-26T10:58:35+00:00","modifiedTime":"2016-03-26T10:58:35+00:00","timestamp":"2022-09-14T17:55:50+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Trigonometry","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33729"},"slug":"trigonometry","categoryId":33729}],"title":"How to Divide a Line Segment into Multiple Parts","strippedTitle":"how to divide a line segment into multiple parts","slug":"how-to-divide-a-line-segment-into-multiple-parts","canonicalUrl":"","seo":{"metaDescription":"If you can find the midpoint of a segment, you can divide it into two equal parts. Finding the middle of each of the two equal parts allows you to find the poin","noIndex":0,"noFollow":0},"content":"<p>If you can find the midpoint of a segment, you can divide it into two equal parts. Finding the middle of each of the two equal parts allows you to find the points needed to divide the entire segment into four equal parts. Finding the middle of each of these segments gives you eight equal parts, and so on.</p>\n<p>For example, to divide the segment with endpoints (–15,10) and (9,2) into eight equal parts, find the various midpoints like so:</p>\n<ul class=\"level-one\">\n <li><p class=\"first-para\">The midpoint of the main segment from (–15,10) to (9,2) is (–3,6).</p>\n </li>\n <li><p class=\"first-para\">The midpoint of half of the main segment, from (–15,10) to (–3,6), is (–9,8), and the midpoint of the other half of the main segment, from (–3,6) to (9,2), is (3,4).</p>\n </li>\n <li><p class=\"first-para\">The midpoints of the four segments determined above are (–12,9), (–6,7), (0,5), and (6,3).</p>\n </li>\n</ul>\n<p>The figure shows the coordinates of the points that divide this line segment into eight equal parts.</p>\n<img src=\"https://sg.cdnki.com/to-divide-a-line-into-two-equal-parts---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzQzODgyMC5pbWFnZTAuanBn.webp\" width=\"535\" height=\"261\" alt=\"image0.jpg\"/>\n<p>Using the midpoint method is fine, as long as you just want to divide a segment into an even number of equal segments. But your job isn't always so easy. For instance, you may need to divide a segment into three equal parts, five equal parts, or some other odd number of equal parts.</p>\n<p>To find a point that isn't equidistant from the endpoints of a segment, just use this formula:</p>\n<img src=\"https://sg.cdnki.com/to-divide-a-line-into-two-equal-parts---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzQzODgyMS5pbWFnZTEuanBn.webp\" width=\"458\" height=\"52\" alt=\"image1.jpg\"/>\n<p>In this formula, (<i>x</i><sub>1</sub>,<i>y</i><sub>1</sub>) is the endpoint where you're starting, (<i>x</i><sub>2</sub>,<i>y</i><sub>2</sub>) is the other endpoint, and <i>k</i> is the fractional part of the segment you want.</p>\n<p>So, to find the coordinates that divide the segment with endpoints (–4,1) and (8,7) into three equal parts, first find the point that's one-third of the distance from (–4,1) to the other endpoint, and then find the point that's two-thirds of the distance from (–4,1) to the other endpoint. The following steps show you how.</p>\n<p>To find the point that's one-third of the distance from (–4,1) to the other endpoint, (8,7):</p>\n<ol class=\"level-one\">\n <li><p class=\"first-para\">Replace <i>x</i><sub>1</sub> with –4, <i>x</i><sub>2</sub> with 8, <i>y</i><sub>1</sub> with 1, <i>y</i><sub>2 </sub>with 7, and <i>k</i> with 1/3.</p>\n<img src=\"https://sg.cdnki.com/to-divide-a-line-into-two-equal-parts---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzQzODgyMi5pbWFnZTIuanBn.webp\" width=\"430\" height=\"72\" alt=\"image2.jpg\"/>\n </li>\n <li><p class=\"first-para\">Subtract the values in the inner parentheses.</p>\n<img src=\"https://sg.cdnki.com/to-divide-a-line-into-two-equal-parts---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzQzODgyMy5pbWFnZTMuanBn.webp\" width=\"264\" height=\"72\" alt=\"image3.jpg\"/>\n </li>\n <li><p class=\"first-para\">Do the multiplication and then add the results to get the coordinates.</p>\n<p class=\"child-para\">=(–4 + 4,1 + 2) = (0,3)</p>\n </li>\n</ol>\n<p>To find the point that's two-thirds of the distance from (–4,1) to the other endpoint, (8,7):</p>\n<ol class=\"level-one\">\n <li><p class=\"first-para\">Replace <i>x</i><sub>1</sub> with –4, <i>x</i><sub>2</sub> with 8, <i>y</i><sub>1</sub> with 1, <i>y</i><sub>2 </sub>with 7, and <i>k</i> with 2/3.</p>\n<img src=\"https://sg.cdnki.com/to-divide-a-line-into-two-equal-parts---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzQzODgyNC5pbWFnZTQuanBn.webp\" width=\"432\" height=\"74\" alt=\"image4.jpg\"/>\n </li>\n <li><p class=\"first-para\">Subtract the values in the inner parentheses.</p>\n<img src=\"https://sg.cdnki.com/to-divide-a-line-into-two-equal-parts---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzQzODgyNS5pbWFnZTUuanBn.webp\" width=\"264\" height=\"74\" alt=\"image5.jpg\"/>\n </li>\n <li><p class=\"first-para\">Do the multiplication and then add the results to get the coordinates.</p>\n<p class=\"child-para\">=(–4 + 8,1 + 4) = (4,5)</p>\n </li>\n</ol>\n<p>The following figure shows the graph of this line segment and the points that divide it into three equal parts.</p>\n<img src=\"https://sg.cdnki.com/to-divide-a-line-into-two-equal-parts---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzQzODgyNi5pbWFnZTYuanBn.webp\" width=\"535\" height=\"279\" alt=\"image6.jpg\"/>","description":"<p>If you can find the midpoint of a segment, you can divide it into two equal parts. Finding the middle of each of the two equal parts allows you to find the points needed to divide the entire segment into four equal parts. Finding the middle of each of these segments gives you eight equal parts, and so on.</p>\n<p>For example, to divide the segment with endpoints (–15,10) and (9,2) into eight equal parts, find the various midpoints like so:</p>\n<ul class=\"level-one\">\n <li><p class=\"first-para\">The midpoint of the main segment from (–15,10) to (9,2) is (–3,6).</p>\n </li>\n <li><p class=\"first-para\">The midpoint of half of the main segment, from (–15,10) to (–3,6), is (–9,8), and the midpoint of the other half of the main segment, from (–3,6) to (9,2), is (3,4).</p>\n </li>\n <li><p class=\"first-para\">The midpoints of the four segments determined above are (–12,9), (–6,7), (0,5), and (6,3).</p>\n </li>\n</ul>\n<p>The figure shows the coordinates of the points that divide this line segment into eight equal parts.</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/438820.image0.jpg\" width=\"535\" height=\"261\" alt=\"image0.jpg\"/>\n<p>Using the midpoint method is fine, as long as you just want to divide a segment into an even number of equal segments. But your job isn't always so easy. For instance, you may need to divide a segment into three equal parts, five equal parts, or some other odd number of equal parts.</p>\n<p>To find a point that isn't equidistant from the endpoints of a segment, just use this formula:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/438821.image1.jpg\" width=\"458\" height=\"52\" alt=\"image1.jpg\"/>\n<p>In this formula, (<i>x</i><sub>1</sub>,<i>y</i><sub>1</sub>) is the endpoint where you're starting, (<i>x</i><sub>2</sub>,<i>y</i><sub>2</sub>) is the other endpoint, and <i>k</i> is the fractional part of the segment you want.</p>\n<p>So, to find the coordinates that divide the segment with endpoints (–4,1) and (8,7) into three equal parts, first find the point that's one-third of the distance from (–4,1) to the other endpoint, and then find the point that's two-thirds of the distance from (–4,1) to the other endpoint. The following steps show you how.</p>\n<p>To find the point that's one-third of the distance from (–4,1) to the other endpoint, (8,7):</p>\n<ol class=\"level-one\">\n <li><p class=\"first-para\">Replace <i>x</i><sub>1</sub> with –4, <i>x</i><sub>2</sub> with 8, <i>y</i><sub>1</sub> with 1, <i>y</i><sub>2 </sub>with 7, and <i>k</i> with 1/3.</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/438822.image2.jpg\" width=\"430\" height=\"72\" alt=\"image2.jpg\"/>\n </li>\n <li><p class=\"first-para\">Subtract the values in the inner parentheses.</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/438823.image3.jpg\" width=\"264\" height=\"72\" alt=\"image3.jpg\"/>\n </li>\n <li><p class=\"first-para\">Do the multiplication and then add the results to get the coordinates.</p>\n<p class=\"child-para\">=(–4 + 4,1 + 2) = (0,3)</p>\n </li>\n</ol>\n<p>To find the point that's two-thirds of the distance from (–4,1) to the other endpoint, (8,7):</p>\n<ol class=\"level-one\">\n <li><p class=\"first-para\">Replace <i>x</i><sub>1</sub> with –4, <i>x</i><sub>2</sub> with 8, <i>y</i><sub>1</sub> with 1, <i>y</i><sub>2 </sub>with 7, and <i>k</i> with 2/3.</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/438824.image4.jpg\" width=\"432\" height=\"74\" alt=\"image4.jpg\"/>\n </li>\n <li><p class=\"first-para\">Subtract the values in the inner parentheses.</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/438825.image5.jpg\" width=\"264\" height=\"74\" alt=\"image5.jpg\"/>\n </li>\n <li><p class=\"first-para\">Do the multiplication and then add the results to get the coordinates.</p>\n<p class=\"child-para\">=(–4 + 8,1 + 4) = (4,5)</p>\n </li>\n</ol>\n<p>The following figure shows the graph of this line segment and the points that divide it into three equal parts.</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/438826.image6.jpg\" width=\"535\" height=\"279\" alt=\"image6.jpg\"/>","blurb":"","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":" <p><b>Mary Jane Sterling</b> is the author of <i>Algebra I For Dummies, Algebra Workbook For Dummies,</i> and many other <i>For Dummies</i> books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics. ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"primaryCategoryTaxonomy":{"categoryId":33729,"title":"Trigonometry","slug":"trigonometry","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33729"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":207754,"title":"Trigonometry For Dummies Cheat Sheet","slug":"trigonometry-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","trigonometry"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/207754"}},{"articleId":203563,"title":"How to Recognize Basic Trig Graphs","slug":"how-to-recognize-basic-trig-graphs","categoryList":["academics-the-arts","math","trigonometry"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/203563"}},{"articleId":203561,"title":"How to Create a Table of Trigonometry Functions","slug":"how-to-create-a-table-of-trigonometry-functions","categoryList":["academics-the-arts","math","trigonometry"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/203561"}},{"articleId":186910,"title":"Comparing Cosine and Sine Functions in a Graph","slug":"comparing-cosine-and-sine-functions-in-a-graph","categoryList":["academics-the-arts","math","trigonometry"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/186910"}},{"articleId":157287,"title":"Signs of Trigonometry Functions in Quadrants","slug":"signs-of-trigonometry-functions-in-quadrants","categoryList":["academics-the-arts","math","trigonometry"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/157287"}}],"fromCategory":[{"articleId":207754,"title":"Trigonometry For Dummies Cheat Sheet","slug":"trigonometry-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","trigonometry"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/207754"}},{"articleId":203563,"title":"How to Recognize Basic Trig Graphs","slug":"how-to-recognize-basic-trig-graphs","categoryList":["academics-the-arts","math","trigonometry"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/203563"}},{"articleId":203561,"title":"How to Create a Table of Trigonometry Functions","slug":"how-to-create-a-table-of-trigonometry-functions","categoryList":["academics-the-arts","math","trigonometry"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/203561"}},{"articleId":199411,"title":"Defining the Radian in Trigonometry","slug":"defining-the-radian-in-trigonometry","categoryList":["academics-the-arts","math","trigonometry"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/199411"}},{"articleId":187511,"title":"How to Use the Double-Angle Identity for Sine","slug":"how-to-use-the-double-angle-identity-for-sine","categoryList":["academics-the-arts","math","trigonometry"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/187511"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282640,"slug":"trigonometry-for-dummies-2nd-edition","isbn":"9781118827413","categoryList":["academics-the-arts","math","trigonometry"],"amazon":{"default":"https://www.amazon.com/gp/product/1118827414/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1118827414/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1118827414-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1118827414/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1118827414/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/trigonometry-for-dummies-2nd-edition-cover-9781118827413-203x255.jpg","width":203,"height":255},"title":"Trigonometry For Dummies","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"<p><b data-author-id=\"8985\">Mary Jane Sterling</b> is the author of <i>Algebra I For Dummies</i> and many other <i>For Dummies</i> titles. She has been teaching mathematics at Bradley University in Peoria, Illinois, for more than 30 years and has loved working with future business executives, physical therapists, teachers, and many others.</p>","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":" <p><b>Mary Jane Sterling</b> is the author of <i>Algebra I For Dummies, Algebra Workbook For Dummies,</i> and many other <i>For Dummies</i> books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics. ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{"key":"cat","values":["academics-the-arts","math","trigonometry"]},{"key":"isbn","values":["9781118827413"]}]\" id=\"du-slot-632215a6c60f7\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{"key":"cat","values":["academics-the-arts","math","trigonometry"]},{"key":"isbn","values":["9781118827413"]}]\" id=\"du-slot-632215a6d2385\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":null,"lifeExpectancySetFrom":null,"dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":149453},"articleLoadedStatus":"success"},"listState":{"list":{},"objectTitle":"","status":"initial","pageType":null,"objectId":null,"page":1,"sortField":"time","sortOrder":1,"categoriesIds":[],"articleTypes":[],"filterData":{},"filterDataLoadedStatus":"initial","pageSize":10},"adsState":{"pageScripts":{"headers":{"timestamp":"2022-09-08T00:59:03+00:00"},"adsId":0,"data":{"scripts":[{"pages":["all"],"location":"header","script":"<!--Optimizely 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If you can find the midpoint of a segment, you can divide it into two equal parts. Finding the middle of each of the two equal parts allows you to find the points needed to divide the entire segment into four equal parts. Finding the middle of each of these segments gives you eight equal parts, and so on. For example, to divide the segment with endpoints (–15,10) and (9,2) into eight equal parts, find the various midpoints like so: The midpoint of the main segment from (–15,10) to (9,2) is (–3,6). The midpoint of half of the main segment, from (–15,10) to (–3,6), is (–9,8), and the midpoint of the other half of the main segment, from (–3,6) to (9,2), is (3,4). The midpoints of the four segments determined above are (–12,9), (–6,7), (0,5), and (6,3). The figure shows the coordinates of the points that divide this line segment into eight equal parts. Using the midpoint method is fine, as long as you just want to divide a segment into an even number of equal segments. But your job isn't always so easy. For instance, you may need to divide a segment into three equal parts, five equal parts, or some other odd number of equal parts. To find a point that isn't equidistant from the endpoints of a segment, just use this formula: In this formula, (x1,y1) is the endpoint where you're starting, (x2,y2) is the other endpoint, and k is the fractional part of the segment you want. So, to find the coordinates that divide the segment with endpoints (–4,1) and (8,7) into three equal parts, first find the point that's one-third of the distance from (–4,1) to the other endpoint, and then find the point that's two-thirds of the distance from (–4,1) to the other endpoint. The following steps show you how. To find the point that's one-third of the distance from (–4,1) to the other endpoint, (8,7):
To find the point that's two-thirds of the distance from (–4,1) to the other endpoint, (8,7):
The following figure shows the graph of this line segment and the points that divide it into three equal parts. |