{"id":142,"date":"2017-08-23T09:08:51","date_gmt":"2017-08-23T09:08:51","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=142"},"modified":"2017-09-18T18:40:22","modified_gmt":"2017-09-18T18:40:22","slug":"the-pythagorean-theorem","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/the-pythagorean-theorem\/","title":{"rendered":"The Pythagorean Theorem"},"content":{"rendered":"<div class=\"twelve columns\" style=\"margin-top: 10%;\">\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/area-of-triangles-and-special-quadrilaterals\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/geometry-spatial-reasoning\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/circles\">Next Lesson \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">The Pythagorean Theorem<\/h1>\n<h4>Objective<\/h4>\n<p>In this lesson, we will prove the Pythagorean Theorem and its converse, and we will prove the formulas for special\u00a0right triangles that were covered in a previous section.<\/p>\n<h4>Previously Covered:<\/h4>\n<ul>\n<li><strong>Side-Side-Side (SSS) postulate : <\/strong>If\u00a0three sides of one triangle are congruent to three sides of\u00a0another triangle, then the two triangles are congruent.<\/li>\n<li><strong><em>Side-Angle-Side(SAS) postulate : <\/em><\/strong>If\u00a0two sides and the included angle of one triangle are congruent to\u00a0two sides and the included angle of another triangle, then the\u00a0two triangles are congruent.<\/li>\n<li>A <strong><em>right triangle\u00a0<\/em><\/strong>is a triangle that includes only one right angle.<\/li>\n<li>In a right triangle, the side\u00a0opposite the right angle is the <em><strong>hypotenuse<\/strong><\/em>,\u00a0and the other two sides are the <strong><em>legs<\/em>. <\/strong><\/li>\n<li>In a 30-60-90 triangle, the\u00a0length of the hypotenuse is twice the length of the shorter leg\u00a0and the length of the longer leg is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s12_p1_clip_image002.gif\" width=\"18\" height=\"18\" name=\"graphics2\" align=\"BOTTOM\" border=\"0\" \/>\u00a0times the shorter leg.<\/li>\n<li>In a 45-45-90 triangle, the\u00a0length of the hypotenuse is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s12_p1_clip_image004.gif\" width=\"19\" height=\"18\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/>\u00a0times the shorter leg.<\/li>\n<li>The area of a triangle is one-half the product of its base\u00a0and its height:\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s12_p1_clip_image006.gif\" width=\"45\" height=\"34\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/li>\n<\/ul>\n<section>\n<h3><strong>How do we prove the Pythagorean Theorem?<\/strong><\/h3>\n<p>One of the pillars of geometry is a well-known theorem created\u00a0by the Greek mathematician Pythagoras:<\/p>\n<p><em><span style=\"text-decoration: none;\">For any right\u00a0triangle with legs a and b and hypotenuse c, the following\u00a0relationship is satisfied: a<sup>2<\/sup> + b<sup>2<\/sup> = c<sup>2<\/sup><br \/>\n<\/span><\/em>In other words, the square of the hypotenuse is equal to the\u00a0sum of the squares of the legs in any right triangle.<\/p>\n<p>The Pythagorean Theorem is a fundamental geometric principle\u00a0that shows up in geometry and trigonometry. We will use it in a\u00a0later section to derive the equation of a circle.<\/p>\n<p>There is more than one way to prove the Pythagorean Theorem,\u00a0but we will only cover only one approach here.<\/p>\n<p><span style=\"text-decoration: none;\">Suppose we are given any\u00a0right triangle with side lengths <\/span><em><span style=\"text-decoration: none;\">a,\u00a0b<\/span><\/em><span style=\"text-decoration: none;\">, and <\/span><em><span style=\"text-decoration: none;\">c<\/span><\/em><span style=\"text-decoration: none;\">,\u00a0where <\/span><em><span style=\"text-decoration: none;\">c\u00a0<\/span><\/em><span style=\"text-decoration: none;\">is the length of the hypotenuse<\/span>.<\/p>\n<p><strong>Step 1:<\/strong> I<span style=\"text-decoration: none;\">f\u00a0we take a square with side length <\/span><em><span style=\"text-decoration: none;\">a\u00a0+ b<\/span><\/em><span style=\"text-decoration: none;\">, we can draw\u00a0four triangles inside the square with legs <\/span><em><span style=\"text-decoration: none;\">a\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and <\/span><em><span style=\"text-decoration: none;\">b\u00a0<\/span><\/em><span style=\"text-decoration: none;\">as shown below. <\/span><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/Math%20Mod%204.1%20Art%20001%20%20Pyth.JPG\" width=\"173\" height=\"177\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p><strong>Step 2:<\/strong> <span style=\"text-decoration: none;\">These\u00a0four triangles are congruent by the SAS postulate. Therefore, they\u00a0all have a hypotenuse of length\u00a0<\/span><em><span style=\"text-decoration: none;\">c<\/span><\/em><span style=\"text-decoration: none;\">. <\/span><\/p>\n<p><strong>Step 3:<\/strong> <span style=\"text-decoration: none;\">The\u00a0quadrilateral formed by the four sides of length <\/span><em><span style=\"text-decoration: none;\">c\u00a0<\/span><\/em><span style=\"text-decoration: none;\">is a square . This follows from the fact that the angles labeled 1\u00a0and 2 add up to 90 \u00b0 , and that one angle of the quadrilateral\u00a0is the supplement of the angles 1 and 2. <\/span><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/Math%20Mod%204.2%20Art%20002%20%20Pyth.JPG\" alt=\"First step in Pythagorean proof\" width=\"173\" height=\"177\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p><strong>Step 4:<\/strong> The area of the larger square is equal\u00a0to the area of the four congruent triangles plus the area of the\u00a0smaller square.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s12_p2_clip_image002.gif\" width=\"216\" height=\"89\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>We can also prove two theorems from the previous section on\u00a0special right triangles using the Pythagorean Theorem.<\/p>\n<p>In a 30-60-90 triangle, the length of the hypotenuse is twice\u00a0the length of the shorter leg and the length of the longer leg is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s12_p2_clip_image004.gif\" width=\"18\" height=\"18\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/>\u00a0times the shorter leg.<\/p>\n<p><span style=\"text-decoration: none;\">If we reflect a 30-60-90\u00a0triangle across its longer side, we obtain an equilateral triangle. Label the sides <\/span><em><span style=\"text-decoration: none;\">a<\/span><\/em><span style=\"text-decoration: none;\">,\u00a0<\/span><em><span style=\"text-decoration: none;\">b<\/span><\/em><span style=\"text-decoration: none;\">,\u00a0and<\/span><em><span style=\"text-decoration: none;\"> c<\/span><\/em><span style=\"text-decoration: none;\">. Thus\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s12_p2_clip_image006.gif\" width=\"43\" height=\"34\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0or equivalently, <\/span><em><span style=\"text-decoration: none;\">c\u00a0= 2b<\/span><\/em><span style=\"text-decoration: none;\">. Find <\/span><em><span style=\"text-decoration: none;\">c\u00a0<\/span><\/em><span style=\"text-decoration: none;\">by using the\u00a0Pythagorean Theorem: <\/span><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s12_p2_clip_image008.gif\" width=\"96\" height=\"126\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<h3>The Pythagorean Theorem<\/h3>\n<p style=\"text-decoration: none;\">In a 45-45-90 triangle, the\u00a0length of the hypotenuse is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s12_p3_clip_image002.gif\" width=\"19\" height=\"18\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/>\u00a0times the shorter leg.\u00a0<span style=\"text-decoration: none;\">Label the length of the two\u00a0congruent sides <\/span><em><span style=\"text-decoration: none;\">a\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and the hypotenuse <\/span><em><span style=\"text-decoration: none;\">c<\/span><\/em><span style=\"text-decoration: none;\">.\u00a0Then by the Pythagorean Theorem, <\/span><\/p>\n<p align=\"CENTER\"><em><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s12_p3_clip_image004.gif\" width=\"90\" height=\"99\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/em><\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>If a rectangle has a diagonal of length 13 and a side of length\u00a05, what is the length of the longer side?<\/p>\n<ol>\n<li>8<\/li>\n<li>10<\/li>\n<li>12<\/li>\n<li>144<\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p><span style=\"text-decoration: none;\">The correct answer is C .\u00a0By the Pythagorean Theorem, we have\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s12_p3_clip_image006.gif\" width=\"79\" height=\"14\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/>,\u00a0where <\/span><em><span style=\"text-decoration: none;\">x\u00a0<\/span><\/em><span style=\"text-decoration: none;\">is the length of the longer side. <\/span><\/p>\n<p><span style=\"text-decoration: none;\">Simplify to find the value\u00a0of <\/span><em><span style=\"text-decoration: none;\">x<\/span><\/em><span style=\"text-decoration: none;\">:\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s12_p3_clip_image008.gif\" width=\"185\" height=\"14\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/><\/span>.<\/p>\n<\/div>\n<\/section>\n<h4><strong>How do we prove the converse of Pythagorean Theorem?<\/strong><\/h4>\n<p>The converse of the Pythagorean Theorem is as follows:<\/p>\n<blockquote><p><em><span style=\"text-decoration: none;\">If the sum of\u00a0the squares of the lengths of two sides of a triangle is equal to\u00a0the square of the length of the third side, then the triangle is a\u00a0right triangle.<\/span><\/em><\/p><\/blockquote>\n<p>To prove the converse, we need to show that one of the angles\u00a0is a right angle . If we show congruence between the given\u00a0triangle and a right triangle, then we will have proved the\u00a0converse .<\/p>\n<p><span style=\"text-decoration: none;\">Suppose we are given a\u00a0triangle angle <\/span><em><span style=\"text-decoration: none;\">ABC\u00a0<\/span><\/em><span style=\"text-decoration: none;\">with sides <\/span><em><span style=\"text-decoration: none;\">a, b<\/span><\/em><span style=\"text-decoration: none;\">,\u00a0and <\/span><em><span style=\"text-decoration: none;\">c\u00a0<\/span><\/em><span style=\"text-decoration: none;\">satisfying <\/span><em><span style=\"text-decoration: none;\">a<\/span><\/em><span style=\"text-decoration: none;\"><sup>2\u00a0<\/sup><\/span><em><span style=\"text-decoration: none;\">+ b<\/span><\/em><span style=\"text-decoration: none;\"><sup>2\u00a0<\/sup><\/span><em><span style=\"text-decoration: none;\">= c<\/span><\/em><span style=\"text-decoration: none;\"><sup>2<\/sup>.<\/span><\/p>\n<p><strong>Step 1:<\/strong> <span style=\"text-decoration: none;\">Take\u00a0a right triangle angle\u00a0<\/span><em><span style=\"text-decoration: none;\">DEF<\/span><\/em><span style=\"text-decoration: none;\"> with legs of\u00a0length <\/span><em><span style=\"text-decoration: none;\">a\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and <\/span><em><span style=\"text-decoration: none;\">b\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and hypotenuse <\/span><em><span style=\"text-decoration: none;\">d<\/span><\/em><span style=\"text-decoration: none;\">.<br \/>\n<\/span><\/p>\n<p><strong>Step 2:<\/strong> <span style=\"text-decoration: none;\">triangle\u00a0<\/span><em><span style=\"text-decoration: none;\">DEF<\/span><\/em><span style=\"text-decoration: none;\"> is a right\u00a0triangle, so <\/span><em><span style=\"text-decoration: none;\">a<\/span><\/em><span style=\"text-decoration: none;\"><sup>2\u00a0<\/sup><\/span><em><span style=\"text-decoration: none;\">+ b<\/span><\/em><span style=\"text-decoration: none;\"> <sup>2\u00a0<\/sup><\/span><em><span style=\"text-decoration: none;\">= d<\/span><\/em><span style=\"text-decoration: none;\"> <sup>2\u00a0<\/sup>by the Pythagorean Theorem.<\/span><\/p>\n<p><strong>Step 3:<\/strong> <span style=\"text-decoration: none;\">By\u00a0the SSS postulate, triangle\u00a0<\/span><em><span style=\"text-decoration: none;\">ABC<\/span><\/em><span style=\"text-decoration: none;\"> and triangle\u00a0<\/span><em><span style=\"text-decoration: none;\">DEF<\/span><\/em><span style=\"text-decoration: none;\"> are congruent. So angle<\/span><em><span style=\"text-decoration: none;\"> ABC\u00a0<\/span><\/em><span style=\"text-decoration: none;\">is a right triangle. <\/span><\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p><span style=\"text-decoration: none;\">In a cube with side length\u00a0<\/span><em><span style=\"text-decoration: none;\">s<\/span><\/em><span style=\"text-decoration: none;\">,\u00a0what is the length of a diagonal <\/span><em><span style=\"text-decoration: none;\">d\u00a0<\/span><\/em><span style=\"text-decoration: none;\">connecting opposite faces as shown in the figure be<\/span>low?<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/Math%20Mod%204.4%20Art%20003.JPG\" width=\"87\" height=\"93\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s12_p3_clip_image010.gif\" width=\"19\" height=\"18\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><em>s<\/em><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s12_p3_clip_image012.gif\" width=\"18\" height=\"18\" name=\"graphics9\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><em>s<\/em><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s12_p3_clip_image013.gif\" width=\"19\" height=\"18\" name=\"graphics10\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s12_p3_clip_image015.gif\" width=\"18\" height=\"18\" name=\"graphics11\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p><span style=\"text-decoration: none;\">The correct answer is B .\u00a0The first step in finding the length of <\/span><em><span style=\"text-decoration: none;\">d\u00a0<\/span><\/em><span style=\"text-decoration: none;\">is finding the length <\/span><em><span style=\"text-decoration: none;\">l\u00a0<\/span><\/em><span style=\"text-decoration: none;\">of a diagonal on a face (see the figure below). <\/span><em><span style=\"text-decoration: none;\">d\u00a0<\/span><\/em><span style=\"text-decoration: none;\">is the hypotenuse of a right triangle with legs <\/span><em><span style=\"text-decoration: none;\">l\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and <\/span><em><span style=\"text-decoration: none;\">s\u00a0<\/span><\/em><span style=\"text-decoration: none;\">. By the Pythagorean Theorem,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s12_p3_clip_image017.gif\" width=\"116\" height=\"21\" name=\"graphics12\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0or\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s12_p3_clip_image019.gif\" width=\"49\" height=\"18\" name=\"graphics13\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0. A second application of the Pythagorean Theorem gives\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s12_p3_clip_image021.gif\" width=\"264\" height=\"24\" name=\"graphics14\" align=\"ABSMIDDLE\" border=\"0\" \/><\/span>.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/Math%20Mod%204.4%20Art%20004.JPG\" alt=\"Cube with diagonal solved\" width=\"87\" height=\"93\" name=\"graphics15\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<\/div>\n<\/section>\n<\/section>\n<p><!-- CONTENT ENDS HERE --><\/p>\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/area-of-triangles-and-special-quadrilaterals\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/geometry-spatial-reasoning\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/circles\">Next Lesson \u27a1<\/a><\/div>\n<p><a class=\"backtotop\" href=\"#title\">Back to Top<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u2b05 Previous Lesson\u00a0Workshop Index\u00a0Next Lesson \u27a1 The Pythagorean Theorem Objective In this lesson, we will prove the Pythagorean Theorem and its converse, and we will prove the formulas for special\u00a0right triangles that were covered in a previous section. Previously Covered: Side-Side-Side (SSS) postulate : If\u00a0three sides of one triangle are congruent to three sides of\u00a0another [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-142","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/142","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/comments?post=142"}],"version-history":[{"count":6,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/142\/revisions"}],"predecessor-version":[{"id":693,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/142\/revisions\/693"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=142"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}