{"id":7796,"date":"2023-03-30T22:39:00","date_gmt":"2023-03-30T17:09:00","guid":{"rendered":"https:\/\/physicscatalyst.com\/article\/?p=7796"},"modified":"2023-05-23T16:24:10","modified_gmt":"2023-05-23T10:54:10","slug":"converse-of-pythagoras-theorem","status":"publish","type":"post","link":"https:\/\/physicscatalyst.com\/article\/converse-of-pythagoras-theorem\/","title":{"rendered":"Converse of Pythagoras theorem"},"content":{"rendered":"\n

What is converse of Pythagoras theorem?<\/h2>\n\n\n\n

In a triangle, if square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right
angle<\/p>\n\n\n\n

\"\"<\/a><\/figure>\n\n\n\n

If $AC^2 = AB^2 + BC^2$
then $\\angle ABC =90^0$<\/p>\n\n\n\n

Proof of converse of Pythagoras theorem<\/h2>\n\n\n\n

Let ABC be the triangle and given<\/p>\n\n\n\n

$AC^2 = AB^2 + BC^2$<\/p>\n\n\n\n

To Prove: $\\angle ABC =90^0$<\/p>\n\n\n\n

Proof:<\/p>\n\n\n\n

Lets us construct a triangle PQR right angled at Q such that PQ = AB and QR = BC<\/p>\n\n\n\n

\"\"<\/a><\/figure>\n\n\n\n

Now in right angle triangle PQR , By Pythagoras theorem<\/p>\n\n\n\n

$PR^2 = PQ^2 + QR^2$
or<\/p>\n\n\n\n

$PR^2 = AB^2 + BC^2$ ( as PQ = AB and QR = BC)<\/p>\n\n\n\n

Therefore<\/p>\n\n\n\n

$PR^2 = AC^2$
PR=AC<\/p>\n\n\n\n

In triangle ABC and PQR<\/p>\n\n\n\n

PQ = AB , QR = BC, PR=AC<\/p>\n\n\n\n

By SSS congruence<\/p>\n\n\n\n

$\\Delta ABC \\cong \\Delta PQR$<\/p>\n\n\n\n

Now from CPCT<\/p>\n\n\n\n

$\\angle B = \\angle Q = 90$<\/p>\n\n\n\n

Hence Proved<\/p>\n\n\n\n

Examples<\/h2>\n\n\n\n

Question 1<\/strong>: Given a triangle with sides of length 6 cm, 8 cm, and 10 cm, is the triangle right-angled?<\/p>\n\n\n\n

Answer<\/strong>:
Check if 10\u00b2 = 6\u00b2 + 8\u00b2.
100 = 36 + 64
100 = 100
Yes, the triangle is right-angled.<\/p>\n\n\n\n

Question 2<\/strong>: A triangle has sides measuring 7 cm, 24 cm, and 25 cm. Determine whether the triangle is right-angled or not.<\/p>\n\n\n\n

Answer<\/strong>:
Check if 25\u00b2 = 7\u00b2 + 24\u00b2.
625 = 49 + 576
625 = 625
Yes, the triangle is right-angled.<\/p>\n\n\n\n

Question 3:<\/strong> Can a triangle with side lengths of 5 cm, 12 cm, and 14 cm be a right-angled triangle? Use the converse of Pythagoras Theorem to justify your answer.<\/p>\n\n\n\n

Answer<\/strong>:
Check if 14\u00b2 = 5\u00b2 + 12\u00b2.
196 = 25 + 144
196 = 169
No, the triangle is not right-angled.<\/p>\n\n\n\n

Question 4<\/strong>:
Determine if a triangle with side lengths of 9 cm, 40 cm, and 41 cm is a right-angled triangle.<\/p>\n\n\n\n

Answer<\/strong>:
Check if 41\u00b2 = 9\u00b2 + 40\u00b2.
1681 = 81 + 1600
1681 = 1681
Yes, the triangle is right-angled.<\/p>\n\n\n\n

Question 5<\/strong>: A triangle has sides of length 15 cm, 20 cm, and 25 cm. Use the converse of Pythagoras Theorem to check if the triangle is right-angled.<\/p>\n\n\n\n

Answer<\/strong>: Check if 25\u00b2 = 15\u00b2 + 20\u00b2.
625 = 225 + 400
625 = 625
Yes, the triangle is right-angled.<\/p>\n\n\n\n

Related Articles<\/strong><\/p>\n\n\n\n

Pythagorean Triples Formula<\/a>
Pythagoras Theorem<\/a>
Similar Triangles Notes<\/a>
<\/p>\n","protected":false},"excerpt":{"rendered":"

What is converse of Pythagoras theorem? In a triangle, if square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a rightangle If $AC^2 = AB^2 + BC^2$then $\\angle ABC =90^0$ Proof of converse of Pythagoras theorem Let ABC be […]<\/p>\n","protected":false},"author":8,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[498],"tags":[],"class_list":["post-7796","post","type-post","status-publish","format-standard","hentry","category-maths"],"yoast_head":"\nConverse of Pythagoras theorem - 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In a triangle, if square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a rightangle If $AC^2 = AB^2 + BC^2$then $\\angle ABC =90^0$ Proof of converse of Pythagoras theorem Let ABC be…","_links":{"self":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/7796","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/users\/8"}],"replies":[{"embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/comments?post=7796"}],"version-history":[{"count":3,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/7796\/revisions"}],"predecessor-version":[{"id":8051,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/7796\/revisions\/8051"}],"wp:attachment":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/media?parent=7796"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/categories?post=7796"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/tags?post=7796"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}