{"id":259433,"date":"2025-07-19T01:50:35","date_gmt":"2025-07-19T01:50:35","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=259433"},"modified":"2025-07-19T01:50:37","modified_gmt":"2025-07-19T01:50:37","slug":"solve-the-triangle","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/19\/solve-the-triangle\/","title":{"rendered":"Solve the triangle."},"content":{"rendered":"\n<p>Solve the triangle. Round answers to the nearest tenth. B = \\boxed{\\text{ } }^\\circ, C = \\boxed{\\text{ } }^\\circ, a = \\boxed{\\text{ } }<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/image-544.png\" alt=\"\" class=\"wp-image-259434\"\/><\/figure>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>Here is a step-by-step solution to solve the triangle.<\/p>\n\n\n\n<p><strong>1. Identify the given information:<\/strong><br>From the image, we have a triangle with the following known values:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Angle A = 96.7\u00b0<\/li>\n\n\n\n<li>Side b (the side opposite angle B) = 10.3 km<\/li>\n\n\n\n<li>Side c (the side opposite angle C) = 12.6 km<\/li>\n<\/ul>\n\n\n\n<p>This is a Side-Angle-Side (SAS) case. We need to find the length of side &#8216;a&#8217; and the measures of angles B and C.<\/p>\n\n\n\n<p><strong>2. Find the length of side &#8216;a&#8217; using the Law of Cosines:<\/strong><br>The Law of Cosines formula is: a\u00b2 = b\u00b2 + c\u00b2 &#8211; 2bc cos(A)<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Substitute the known values into the formula:<br>a\u00b2 = (10.3)\u00b2 + (12.6)\u00b2 &#8211; 2(10.3)(12.6)cos(96.7\u00b0)<\/li>\n\n\n\n<li>Calculate the squares and the product:<br>a\u00b2 = 106.09 + 158.76 &#8211; 259.56 * cos(96.7\u00b0)<\/li>\n\n\n\n<li>Calculate the value of cos(96.7\u00b0):<br>cos(96.7\u00b0) \u2248 -0.1167<\/li>\n\n\n\n<li>Continue the calculation for a\u00b2:<br>a\u00b2 \u2248 264.85 &#8211; 259.56 * (-0.1167)<br>a\u00b2 \u2248 264.85 + 30.28<br>a\u00b2 \u2248 295.13<\/li>\n\n\n\n<li>Take the square root to find &#8216;a&#8217;:<br>a = \u221a295.13 \u2248 17.179<\/li>\n\n\n\n<li>Round to the nearest tenth:<br><strong>a = 17.2 km<\/strong><\/li>\n<\/ul>\n\n\n\n<p><strong>3. Find the measure of angle B using the Law of Sines:<\/strong><br>The Law of Sines formula is: a\/sin(A) = b\/sin(B)<br>We can rearrange this to solve for sin(B): sin(B) = (b * sin(A)) \/ a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Substitute the known and calculated values (using the more precise value for &#8216;a&#8217; for better accuracy):<br>sin(B) = (10.3 * sin(96.7\u00b0)) \/ 17.179<\/li>\n\n\n\n<li>Calculate the value of sin(96.7\u00b0):<br>sin(96.7\u00b0) \u2248 0.9932<\/li>\n\n\n\n<li>Continue the calculation for sin(B):<br>sin(B) \u2248 (10.3 * 0.9932) \/ 17.179<br>sin(B) \u2248 10.22996 \/ 17.179<br>sin(B) \u2248 0.5955<\/li>\n\n\n\n<li>Find angle B by taking the inverse sine (arcsin):<br>B = arcsin(0.5955) \u2248 36.55\u00b0<\/li>\n\n\n\n<li>Round to the nearest tenth:<br><strong>B = 36.6\u00b0<\/strong><\/li>\n<\/ul>\n\n\n\n<p><strong>4. Find the measure of angle C:<\/strong><br>The sum of the angles in a triangle is 180\u00b0.<br>C = 180\u00b0 &#8211; A &#8211; B<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Substitute the values for A and B:<br>C = 180\u00b0 &#8211; 96.7\u00b0 &#8211; 36.6\u00b0<br>C = 46.7\u00b0<\/li>\n\n\n\n<li><strong>C = 46.7\u00b0<\/strong><\/li>\n<\/ul>\n\n\n\n<p><strong>Final Answer:<\/strong><br>The solved values for the triangle, rounded to the nearest tenth, are:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>B = 36.6\u00b0<\/strong><\/li>\n\n\n\n<li><strong>C = 46.7\u00b0<\/strong><\/li>\n\n\n\n<li><strong>a = 17.2<\/strong>\u00a0km<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner6-965.jpeg\" alt=\"\" class=\"wp-image-259435\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Solve the triangle. Round answers to the nearest tenth. B = \\boxed{\\text{ } }^\\circ, C = \\boxed{\\text{ } }^\\circ, a = \\boxed{\\text{ } } The Correct Answer and Explanation is: Here is a step-by-step solution to solve the triangle. 1. Identify the given information:From the image, we have a triangle with the following known values: [&hellip;]<\/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-259433","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/259433","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=259433"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/259433\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=259433"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=259433"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=259433"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}