{"id":37971,"date":"2025-06-26T07:13:48","date_gmt":"2025-06-26T07:13:48","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=37971"},"modified":"2025-06-26T07:13:50","modified_gmt":"2025-06-26T07:13:50","slug":"rotate-the-triangle-90-degrees-counter-clockwise-around-the-point-1-1","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/rotate-the-triangle-90-degrees-counter-clockwise-around-the-point-1-1\/","title":{"rendered":"Rotate the triangle 90 degrees counter-clockwise around the point (-1, -1)."},"content":{"rendered":"\n<p>Rotate the triangle 90 degrees counter-clockwise around the point (-1, -1).<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"700\" height=\"247\" src=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-358.png\" alt=\"\" class=\"wp-image-37974\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-358.png 700w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-358-300x106.png 300w\" sizes=\"auto, (max-width: 700px) 100vw, 700px\" \/><\/figure>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To rotate the triangle 90 degrees counter-clockwise about the point (-1, -1), we follow a three-step process: translate, rotate, and translate back.<\/p>\n\n\n\n<p><strong>Step 1: Identify the original coordinates<\/strong> From the image, the triangle has vertices: A(0, 0), B(2, 0), and C(1, 2) The center of rotation is P(-1, -1)<\/p>\n\n\n\n<p><strong>Step 2: Translate the triangle so that P becomes the origin<\/strong> We subtract (-1, -1) from each point:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>A&#8217; = (0 &#8211; (-1), 0 &#8211; (-1)) = (1, 1)<\/li>\n\n\n\n<li>B&#8217; = (2 &#8211; (-1), 0 &#8211; (-1)) = (3, 1)<\/li>\n\n\n\n<li>C&#8217; = (1 &#8211; (-1), 2 &#8211; (-1)) = (2, 3)<\/li>\n<\/ul>\n\n\n\n<p><strong>Step 3: Apply the 90-degree counter-clockwise rotation<\/strong> The rule is (x, y) becomes (-y, x):<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>A&#8221; = (-1, 1)<\/li>\n\n\n\n<li>B&#8221; = (-1, 3)<\/li>\n\n\n\n<li>C&#8221; = (-3, 2)<\/li>\n<\/ul>\n\n\n\n<p><strong>Step 4: Translate the triangle back by adding (-1, -1)<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>A_final = (-1 + (-1), 1 + (-1)) = (-2, 0)<\/li>\n\n\n\n<li>B_final = (-1 + (-1), 3 + (-1)) = (-2, 2)<\/li>\n\n\n\n<li>C_final = (-3 + (-1), 2 + (-1)) = (-4, 1)<\/li>\n<\/ul>\n\n\n\n<p><strong>Final Answer:<\/strong> The rotated triangle has vertices A&#8221; = (-2, 0), B&#8221; = (-2, 2), C&#8221; = (-4, 1)<\/p>\n\n\n\n<p><strong>Explanation<\/strong><\/p>\n\n\n\n<p>Rotating a figure about a point that is not the origin requires two translations. The triangle is first shifted so that the center of rotation, here (-1, -1), becomes the origin. This makes the math easier because the standard rotation rules are defined about the origin.<\/p>\n\n\n\n<p>By translating each vertex, we realign the triangle with respect to the origin. After translation, each point is rotated using the rule for 90 degrees counter-clockwise rotation. In this transformation, each point (x, y) is mapped to (-y, x). This rule comes from how axes are oriented: a 90-degree turn shifts the x-axis to where the y-axis was and the y-axis to where the negative x-axis was.<\/p>\n\n\n\n<p>Once the rotation is complete in this temporary space, we undo the initial translation. By adding the original shift vector (-1, -1), we reposition the triangle to its correct location on the grid. This final triangle represents the image of the original triangle under the described rotation.<\/p>\n\n\n\n<p>This method ensures each transformation is logically sequenced and mathematically accurate. It can be generalized to any point of rotation and is particularly useful when dealing with figures on a coordinate plane.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"852\" height=\"1024\" src=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-992.jpeg\" alt=\"\" class=\"wp-image-37976\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-992.jpeg 852w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-992-250x300.jpeg 250w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-992-768x923.jpeg 768w\" sizes=\"auto, (max-width: 852px) 100vw, 852px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Rotate the triangle 90 degrees counter-clockwise around the point (-1, -1). The Correct Answer and Explanation is: To rotate the triangle 90 degrees counter-clockwise about the point (-1, -1), we follow a three-step process: translate, rotate, and translate back. Step 1: Identify the original coordinates From the image, the triangle has vertices: A(0, 0), B(2, [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-37971","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/37971","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=37971"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/37971\/revisions"}],"predecessor-version":[{"id":37977,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/37971\/revisions\/37977"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=37971"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=37971"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=37971"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}