{"id":196419,"date":"2025-03-05T04:02:37","date_gmt":"2025-03-05T04:02:37","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=196419"},"modified":"2025-03-05T04:02:40","modified_gmt":"2025-03-05T04:02:40","slug":"the-following-two-column-proof-with-missing-justifications-proves-the-pythagorean","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/03\/05\/the-following-two-column-proof-with-missing-justifications-proves-the-pythagorean\/","title":{"rendered":"The following two column proof with missing justifications proves the Pythagorean"},"content":{"rendered":"\n

The following two column proof with missing justifications proves the Pythagorean? Draw an altitude from point C to AB^- Let BC^- = a CA^- = b AB^- = c CD^- = h DB^- = y AD^- = x y – x = c c\/a = a\/y; c\/b = b\/x a^2 = cy; b^2 = cx a^2 + b^2 = cy + cx a^2 + b^2 = c(y + x) a^2 + b^2 = c(c) a^2 + b^2 = c^2 Which is not a justification for the proof? Pieces of Right Triangles Similarity Theorem Side Side Side Similarity Theorem Substitution Addition Property of Equality<\/p>\n\n\n\n

The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n

The missing justification in the proof provided is “Side Side Side Similarity Theorem”<\/strong>.<\/p>\n\n\n\n

Explanation:<\/h3>\n\n\n\n

In the given two-column proof, the goal is to prove the Pythagorean theorem using the concept of right triangles and the relationships between the sides. Let’s break down the justification process:<\/p>\n\n\n\n

    \n
  1. Draw an altitude from point C to AB<\/strong>: This step creates two right triangles, triangle ABC<\/strong> and triangle ACD<\/strong>, which will help establish relationships between the sides.<\/li>\n\n\n\n
  2. BC = a, CA = b, AB = c, CD = h, DB = y, AD = x<\/strong>: These are the given lengths of the sides and the parts of the triangles.<\/li>\n\n\n\n
  3. y – x = c<\/strong>: This equation represents the relationship between the segments formed by the altitude.<\/li>\n\n\n\n
  4. c\/a = a\/y and c\/b = b\/x<\/strong>: These two ratios are based on the similarity of the triangles. Since the triangles are right triangles and share angle C<\/strong>, they are similar by the AA (Angle-Angle) similarity criterion<\/strong>, not by Side Side Side (SSS) similarity<\/strong>. This is a crucial distinction: similarity of triangles involves having the same angle measures, and proportionality of corresponding sides, not just matching side lengths.<\/li>\n\n\n\n
  5. a^2 = cy, b^2 = cx<\/strong>: These equations follow from applying the geometric mean (altitude) theorem, not from SSS similarity.<\/li>\n\n\n\n
  6. a^2 + b^2 = cy + cx<\/strong>: Here, addition of the above equations is justified by the Addition Property of Equality<\/strong>.<\/li>\n\n\n\n
  7. a^2 + b^2 = c(y + x)<\/strong>: Using substitution, we know that (y + x = c) from earlier, so this equation is valid.<\/li>\n\n\n\n
  8. a^2 + b^2 = c^2<\/strong>: Finally, the proof concludes by simplifying the equation, thus proving the Pythagorean theorem.<\/li>\n<\/ol>\n\n\n\n

    Why “Side Side Side Similarity Theorem” is Incorrect:<\/h3>\n\n\n\n
      \n
    • The Side Side Side (SSS) Similarity Theorem<\/strong> is used when two triangles have corresponding sides in the same ratio. However, in this proof, we are using the Angle-Angle (AA) similarity criterion<\/strong> to establish that the triangles are similar. Therefore, the SSS similarity<\/strong> does not apply here.<\/li>\n<\/ul>\n\n\n\n

      The other justifications, such as Substitution<\/strong>, Addition Property of Equality<\/strong>, and the Pieces of Right Triangles Similarity Theorem<\/strong>, are correct and appropriately applied in the proof.<\/p>\n\n\n\n

      Image:<\/h3>\n\n\n\n

      Here is an illustration of the two right triangles (ABC and ACD) with the altitude from C to AB to visualize the relationships between the sides.<\/p>\n\n\n\n

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

      Here is a geometric diagram showing the right triangle ABC<\/strong> with the altitude CD<\/strong> drawn from point C<\/strong> to side AB<\/strong>. The sides are labeled as follows: BC = a<\/strong>, CA = b<\/strong>, AB = c<\/strong>, CD = h<\/strong>, DB = y<\/strong>, and AD = x<\/strong>. The diagram also demonstrates that y + x = c<\/strong>, with the triangles being similar by AA similarity<\/strong>, not SSS similarity<\/strong>, as well as showing the key relationship a^2 + b^2 = c^2<\/strong>.<\/p>\n","protected":false},"excerpt":{"rendered":"

      The following two column proof with missing justifications proves the Pythagorean? Draw an altitude from point C to AB^- Let BC^- = a CA^- = b AB^- = c CD^- = h DB^- = y AD^- = x y – x = c c\/a = a\/y; c\/b = b\/x a^2 = cy; b^2 = cx […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"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":"","ast-disable-related-posts":"","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":[25],"tags":[],"class_list":["post-196419","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/196419","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=196419"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/196419\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=196419"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=196419"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=196419"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}