{"id":85068,"date":"2022-05-31T20:41:34","date_gmt":"2022-05-31T15:11:34","guid":{"rendered":"https:\/\/ccssanswers.com\/?p=85068"},"modified":"2022-05-10T14:19:42","modified_gmt":"2022-05-10T08:49:42","slug":"eureka-math-grade-8-module-4-lesson-31","status":"publish","type":"post","link":"https:\/\/ccssanswers.com\/eureka-math-grade-8-module-4-lesson-31\/","title":{"rendered":"Eureka Math Grade 8 Module 4 Lesson 31 Answer Key"},"content":{"rendered":"<h2>Engage NY Eureka Math 8th Grade Module 4 Lesson 31 Answer Key<\/h2>\n<h3>Eureka Math Grade 8 Module 4 Lesson 31 Exercise Answer Key<\/h3>\n<p>Exercises<\/p>\n<p>Exercise 1.<br \/>\nIdentify two Pythagorean triples using the known triple 3, 4, 5 (other than 6, 8, 10).<br \/>\nAnswer:<br \/>\nAnswers will vary. Accept any triple that is a whole number multiple of 3, 4, 5.<\/p>\n<p>Exercise 2.<br \/>\nIdentify two Pythagorean triples using the known triple 5, 12, 13.<br \/>\nAnswer:<br \/>\nAnswers will vary. Accept any triple that is a whole number multiple of 5, 12, 13.<\/p>\n<p>Exercise 3.<br \/>\nIdentify two triples using either 3, 4, 5 or 5, 12, 13.<br \/>\nAnswer:<br \/>\nAnswers will vary.<\/p>\n<p>Use the system <img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-85070\" src=\"https:\/\/ccssanswers.com\/wp-content\/uploads\/2021\/04\/Engage-NY-Math-Grade-8-Module-4-Lesson-31-Exercise-Answer-Key-1.png\" alt=\"Engage NY Math Grade 8 Module 4 Lesson 31 Exercise Answer Key 1\" width=\"101\" height=\"94\" \/> to find Pythagorean triples for the given values of s and t. Recall that the solution in the form of (\\(\\frac{c}{b}\\), \\(\\frac{a}{b}\\)) is the triple a, b, c.<br \/>\nExercise 4.<br \/>\ns = 4, t = 5<br \/>\nAnswer:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-85071\" src=\"https:\/\/ccssanswers.com\/wp-content\/uploads\/2021\/04\/Engage-NY-Math-Grade-8-Module-4-Lesson-31-Exercise-Answer-Key-2.png\" alt=\"Engage NY Math Grade 8 Module 4 Lesson 31 Exercise Answer Key 2\" width=\"109\" height=\"94\" \/><br \/>\nx + y + x &#8211; y = \\(\\frac{5}{4}\\) + \\(\\frac{4}{5}\\)<br \/>\n2x = \\(\\frac{5}{4}\\) + \\(\\frac{4}{5}\\)<br \/>\n2x = \\(\\frac{41}{20}\\)<br \/>\nx = \\(\\frac{41}{40}\\)<\/p>\n<p>\\(\\frac{41}{40}\\) + y = \\(\\frac{5}{4}\\)<br \/>\ny = \\(\\frac{5}{4}\\) &#8211; \\(\\frac{41}{40}\\)<br \/>\ny = \\(\\frac{9}{40}\\)<\/p>\n<p>Then the solution is (\\(\\frac{41}{40}\\), \\(\\frac{9}{40}\\)), and the triple is 9, 40, 41.<\/p>\n<p>Exercise 5.<br \/>\ns = 7, t = 10<br \/>\nAnswer:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-85072\" src=\"https:\/\/ccssanswers.com\/wp-content\/uploads\/2021\/04\/Engage-NY-Math-Grade-8-Module-4-Lesson-31-Exercise-Answer-Key-3.png\" alt=\"Engage NY Math Grade 8 Module 4 Lesson 31 Exercise Answer Key 3\" width=\"112\" height=\"103\" \/><br \/>\nx + y + x &#8211; y = \\(\\frac{10}{7}\\) + \\(\\frac{7}{10}\\)<br \/>\n2x = \\(\\frac{149}{70}\\)<br \/>\nx = \\(\\frac{149}{140}\\)<\/p>\n<p>\\(\\frac{149}{140}\\) + y = \\(\\frac{10}{7}\\)<br \/>\ny = \\(\\frac{10}{7}\\) &#8211; \\(\\frac{149}{140}\\)<br \/>\ny = \\(\\frac{51}{140}\\)<br \/>\nThen the solution is (\\(\\frac{149}{140}\\), \\(\\frac{51}{140}\\)), and the triple is 51, 140, 149.<\/p>\n<p>Exercise 6.<br \/>\ns = 1, t = 4<br \/>\nAnswer:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-85073\" src=\"https:\/\/ccssanswers.com\/wp-content\/uploads\/2021\/04\/Engage-NY-Math-Grade-8-Module-4-Lesson-31-Exercise-Answer-Key-4.png\" alt=\"Engage NY Math Grade 8 Module 4 Lesson 31 Exercise Answer Key 4\" width=\"107\" height=\"102\" \/><br \/>\nx + y + x &#8211; y = 4 + \\(\\frac{1}{4}\\)<br \/>\n2x = \\(\\frac{17}{4}\\)<br \/>\nx = \\(\\frac{17}{8}\\)<\/p>\n<p>\\(\\frac{17}{8}\\) + y = \\(\\frac{4}{1}\\)<br \/>\ny = 4 &#8211; \\(\\frac{17}{8}\\)<br \/>\ny = \\(\\frac{15}{8}\\)<br \/>\nThen the solution is (\\(\\frac{17}{8}\\), \\(\\frac{15}{8}\\)), and the triple is 15, 8, 17.<\/p>\n<p>Exercise 7.<br \/>\nUse a calculator to verify that you found a Pythagorean triple in each of the Exercises 4\u20136. Show your work below.<br \/>\nAnswer:<br \/>\nFor the triple 9, 40, 41:<br \/>\n9<sup>2<\/sup> + 40<sup>2<\/sup> = 41<sup>2<\/sup><br \/>\n81 + 1600 = 1681<br \/>\n1681 = 1681<\/p>\n<p>For the triple 51, 140, 149:<br \/>\n51<sup>2<\/sup> + 140<sup>2<\/sup> = 149<sup>2<\/sup><br \/>\n2601 + 19600 = 22201<br \/>\n22201 = 22201<\/p>\n<p>For the triple 15, 8, 17:<br \/>\n15<sup>2<\/sup> + 8<sup>2<\/sup> = 17<sup>2<\/sup><br \/>\n225 + 64 = 289<br \/>\n289 = 289<\/p>\n<h3>Eureka Math Grade 8 Module 4 Lesson 31 Problem Set Answer Key<\/h3>\n<p>Question 1.<br \/>\nExplain in terms of similar triangles why it is that when you multiply the known Pythagorean triple 3, 4, 5 by 12, it generates a Pythagorean triple.<br \/>\nAnswer:<br \/>\nThe triangle with lengths 3, 4, 5 is similar to the triangle with lengths 36, 48, 60. They are both right triangles whose corresponding side lengths are equal to the same constant.<br \/>\n\\(\\frac{36}{3}\\) = \\(\\frac{48}{4}\\) = \\(\\frac{60}{5}\\) = 12<br \/>\nTherefore, the triangles are similar, and we can say that there is a dilation from some center with scale factor r = 12 that makes the triangles congruent.<\/p>\n<p>Question 2.<br \/>\nIdentify three Pythagorean triples using the known triple 8, 15, 17.<br \/>\nAnswer:<br \/>\nAnswers will vary. Accept any triple that is a whole number multiple of 8, 15, 17.<\/p>\n<p>Question 3.<br \/>\nIdentify three triples (numbers that satisfy a<sup>2<\/sup> + b<sup>2<\/sup> = c<sup>2<\/sup>, but a, b, c are not whole numbers) using the triple 8, 15, 17.<br \/>\nAnswer:<br \/>\nAnswers will vary. Accept any triple that is not a set of whole numbers.<\/p>\n<p>Use the system <img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-85074\" src=\"https:\/\/ccssanswers.com\/wp-content\/uploads\/2021\/04\/Eureka-Math-8th-Grade-Module-4-Lesson-31-Problem-Set-Answer-Key-1.png\" alt=\"Eureka Math 8th Grade Module 4 Lesson 31 Problem Set Answer Key 1\" width=\"100\" height=\"100\" \/> to find Pythagorean triples for the given values of s and t. Recall that the solution, in the form of (c\/b, a\/b), is the triple, a, b, c.<br \/>\nQuestion 4.<br \/>\ns = 2, t = 9<br \/>\nAnswer:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-85075\" src=\"https:\/\/ccssanswers.com\/wp-content\/uploads\/2021\/04\/Eureka-Math-8th-Grade-Module-4-Lesson-31-Problem-Set-Answer-Key-2.png\" alt=\"Eureka Math 8th Grade Module 4 Lesson 31 Problem Set Answer Key 2\" width=\"100\" height=\"103\" \/><br \/>\nx + y + x &#8211; y = \\(\\frac{9}{2}\\) + \\(\\frac{2}{9}\\)<br \/>\n2x = \\(\\frac{85}{18}\\)<br \/>\nx = \\(\\frac{85}{36}\\)<\/p>\n<p>\\(\\frac{85}{36}\\) + y = \\(\\frac{9}{2}\\)<br \/>\ny = \\(\\frac{9}{2}\\) &#8211; \\(\\frac{85}{36}\\)<br \/>\ny = \\(\\frac{77}{36}\\)<br \/>\nThen the solution is (\\(\\frac{85}{36}\\), \\(\\frac{77}{36}\\)), and the triple is 77, 36, 85.<\/p>\n<p>Question 5.<br \/>\ns = 6, t = 7<br \/>\nAnswer:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-85076\" src=\"https:\/\/ccssanswers.com\/wp-content\/uploads\/2021\/04\/Eureka-Math-8th-Grade-Module-4-Lesson-31-Problem-Set-Answer-Key-3.png\" alt=\"Eureka Math 8th Grade Module 4 Lesson 31 Problem Set Answer Key 3\" width=\"105\" height=\"107\" \/><br \/>\nx + y + x &#8211; y = \\(\\frac{7}{6}\\) + 6\/7<br \/>\n2x = \\(\\frac{85}{42}\\)<br \/>\nx = \\(\\frac{85}{84}\\)<\/p>\n<p>\\(\\frac{85}{84}\\) + y = \\(\\frac{7}{6}\\)<br \/>\ny = \\(\\frac{7}{6}\\) &#8211; \\(\\frac{85}{84}\\)<br \/>\ny = \\(\\frac{13}{84}\\)<br \/>\nThen the solution is (\\(\\frac{85}{84}\\), \\(\\frac{13}{84}\\)), and the triple is 13, 84, 85.<\/p>\n<p>Question 6.<br \/>\ns = 3, t = 4<br \/>\nAnswer:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-85077\" src=\"https:\/\/ccssanswers.com\/wp-content\/uploads\/2021\/04\/Eureka-Math-8th-Grade-Module-4-Lesson-31-Problem-Set-Answer-Key-4.png\" alt=\"Eureka Math 8th Grade Module 4 Lesson 31 Problem Set Answer Key 4\" width=\"113\" height=\"107\" \/><br \/>\nx + y + x &#8211; y = \\(\\frac{4}{3}\\) + \\(\\frac{3}{4}\\)<br \/>\n2x = \\(\\frac{25}{12}\\)<br \/>\nx = \\(\\frac{25}{24}\\)<\/p>\n<p>\\(\\frac{25}{24}\\) + y = \\(\\frac{4}{3}\\)<br \/>\ny = \\(\\frac{4}{3}\\) &#8211; \\(\\frac{25}{24}\\)<br \/>\ny = \\(\\frac{7}{24}\\)<br \/>\nThen the solution is (\\(\\frac{25}{24}\\), \\(\\frac{7}{24}\\)), and the triple is 7, 24, 25.<\/p>\n<p>Question 7.<br \/>\nUse a calculator to verify that you found a Pythagorean triple in each of the Problems 4\u20136. Show your work.<br \/>\nAnswer:<br \/>\nFor the triple 77, 36, 85:<br \/>\n77<sup>2<\/sup> + 36<sup>2<\/sup> = 85<sup>2<\/sup><br \/>\n5929 + 1296 = 7225<br \/>\n7225 = 7225<\/p>\n<p>For the triple 13, 84, 85:<br \/>\n13<sup>2<\/sup> + 84<sup>2<\/sup> = 85<sup>2<\/sup><br \/>\n169 + 7056 = 7225<br \/>\n7225 = 7225<\/p>\n<p>For the triple 7, 24, 25:<br \/>\n7<sup>2<\/sup> + 24<sup>2<\/sup> = 25<sup>2<\/sup><br \/>\n49 + 576 = 625<br \/>\n625 = 625<\/p>\n<h3>Eureka Math Grade 8 Module 4 Lesson 31 Exit Ticket Answer Key<\/h3>\n<p>Use a calculator to complete Problems 1\u20133.<br \/>\nQuestion 1.<br \/>\nIs 7, 20, 21 a Pythagorean triple? Is 1, \\(\\frac{15}{8}\\), \\(\\frac{17}{8}\\) a Pythagorean triple? Explain.<br \/>\nAnswer:<br \/>\nThe set of numbers 7, 20, 21 is not a Pythagorean triple because 7<sup>2<\/sup> + 20<sup>2<\/sup> \u2260 21<sup>2<\/sup>.<br \/>\nThe set of numbers 1, \\(\\frac{15}{8}\\), \\(\\frac{17}{8}\\) is not a Pythagorean triple because the numbers \\(\\frac{15}{8}\\) and \\(\\frac{17}{8}\\) are not whole numbers.<br \/>\nBut they are a triple because 1<sup>2<\/sup> + (\\(\\frac{15}{8}\\))<sup>2<\/sup> = (\\(\\frac{17}{8}\\))<sup>2<\/sup>.<\/p>\n<p>Question 2.<br \/>\nIdentify two Pythagorean triples using the known triple 9, 40, 41.<br \/>\nAnswer:<br \/>\nAnswers will vary. Accept any triple that is a whole number multiple of 9, 40, 41.<\/p>\n<p>Question 3.<br \/>\nUse the system <img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-85078\" src=\"https:\/\/ccssanswers.com\/wp-content\/uploads\/2021\/04\/Eureka-Math-Grade-8-Module-4-Lesson-31-Exit-Ticket-Answer-Key-1.png\" alt=\"Eureka Math Grade 8 Module 4 Lesson 31 Exit Ticket Answer Key 1\" width=\"86\" height=\"77\" \/> to find Pythagorean triples for the given values of s = 2 and t = 3. Recall that the solution in the form of (\\(\\frac{c}{b}\\), \\(\\frac{a}{b}\\)) is the triple a, b, c. Verify your results.<br \/>\nAnswer:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-85079\" src=\"https:\/\/ccssanswers.com\/wp-content\/uploads\/2021\/04\/Eureka-Math-Grade-8-Module-4-Lesson-31-Exit-Ticket-Answer-Key-2.png\" alt=\"Eureka Math Grade 8 Module 4 Lesson 31 Exit Ticket Answer Key 2\" width=\"107\" height=\"102\" \/><br \/>\nx + y + x &#8211; y = \\(\\frac{3}{2}\\) + \\(\\frac{2}{3}\\)<br \/>\n2x = \\(\\frac{13}{6}\\)<br \/>\nx = \\(\\frac{13}{12}\\)<\/p>\n<p>\\(\\frac{13}{12}\\) + y = \\(\\frac{3}{2}\\)<br \/>\ny = \\(\\frac{3}{2}\\) &#8211; \\(\\frac{13}{12}\\)<br \/>\ny = \\(\\frac{5}{12}\\)<br \/>\nThen the solution is (\\(\\frac{13}{12}\\), \\(\\frac{5}{12}\\)), and the triple is 5, 12, 13.<br \/>\n5<sup>2<\/sup> + 12<sup>2<\/sup> = 13<sup>2<\/sup><br \/>\n25 + 144 = 169<br \/>\n169 = 169<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Engage NY Eureka Math 8th Grade Module 4 Lesson 31 Answer Key Eureka Math Grade 8 Module 4 Lesson 31 Exercise Answer Key Exercises Exercise 1. Identify two Pythagorean triples using the known triple 3, 4, 5 (other than 6, 8, 10). Answer: Answers will vary. Accept any triple that is a whole number multiple &#8230; <a title=\"Eureka Math Grade 8 Module 4 Lesson 31 Answer Key\" class=\"read-more\" href=\"https:\/\/ccssanswers.com\/eureka-math-grade-8-module-4-lesson-31\/\" aria-label=\"More on Eureka Math Grade 8 Module 4 Lesson 31 Answer Key\">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":[6],"tags":[],"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/posts\/85068"}],"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=85068"}],"version-history":[{"count":1,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/posts\/85068\/revisions"}],"predecessor-version":[{"id":137236,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/posts\/85068\/revisions\/137236"}],"wp:attachment":[{"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/media?parent=85068"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/categories?post=85068"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/tags?post=85068"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}