{"id":44208,"date":"2023-05-24T15:35:03","date_gmt":"2023-05-24T10:05:03","guid":{"rendered":"https:\/\/www.practicemock.com\/blog\/?p=44208"},"modified":"2023-05-24T15:35:03","modified_gmt":"2023-05-24T10:05:03","slug":"ssc-cgl-2023-important-quant-topics-triangle","status":"publish","type":"post","link":"https:\/\/www.practicemock.com\/blog\/ssc-cgl-2023-important-quant-topics-triangle\/","title":{"rendered":"SSC CGL 2023- Important Quant Topics (Triangle)"},"content":{"rendered":"\n<p>In SSC CGL exam, several questions are asked in Geometry. In this article, we will run you through several concepts related to Geometry along with examples so that you can understand them easily. Let\u2019s get started.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Incircle of a triangle<\/h2>\n\n\n\n<p>An incircle is a circle that is tangent to all three sides of a triangle. The center of the incircle is called the incenter. The incenter can be found by intersecting the angle bisectors of the triangle.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.practicemock.com\/blog\/wp-content\/uploads\/2023\/05\/Incircle-1.png\" alt=\"\" class=\"wp-image-44210\" width=\"269\" height=\"254\"\/><\/figure>\n\n\n\n<p>The inradius is the radius of the incircle. The inradius can be found by dividing the area of the triangle by the semiperimeter of the triangle.<\/p>\n\n\n\n<p>The incircle has several properties, including:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>The inradius is equal to the ratio of the area of the triangle to the semiperimeter of the triangle.<\/li><li>The incenter is equidistant from the three sides of the triangle.<\/li><li>The incircle is tangent to the three sides of the triangle at the midpoints of the sides.<\/li><li>The incircle is tangent to the nine-point circle of the triangle.<\/li><\/ul>\n\n\n\n<p>The incircle can be used to solve a variety of problems involving triangles, such as finding the area of the triangle, finding the length of a side of the triangle, or finding the distance between two points on a side of the triangle.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><a href=\"https:\/\/www.practicemock.com\/ssc-cgl\/?ref=7334\"><img loading=\"lazy\" decoding=\"async\" width=\"3000\" height=\"375\" src=\"https:\/\/www.practicemock.com\/blog\/wp-content\/uploads\/2023\/05\/SSC-CGL-Free-Mock-2-1.png\" alt=\"\" class=\"wp-image-44253\"\/><\/a><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">Inradius of a triangle<\/h2>\n\n\n\n<p>An inradius is the radius of a circle that is tangent to all three sides of a triangle. The inradius is denoted by r.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"357\" height=\"243\" src=\"https:\/\/www.practicemock.com\/blog\/wp-content\/uploads\/2023\/05\/Inradius-1.png\" alt=\"\" class=\"wp-image-44211\"\/><\/figure>\n\n\n\n<p>The inradius can be found using the following formula:<\/p>\n\n\n\n<p>r = S \/ s<\/p>\n\n\n\n<p>where S is the area of the triangle and s is the semiperimeter of the triangle.<\/p>\n\n\n\n<p>The inradius has several properties, including:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>The inradius is equal to the ratio of the area of the triangle to the semiperimeter of the triangle.<\/li><li>The incenter is equidistant from the three sides of the triangle.<\/li><li>The incircle is tangent to the three sides of the triangle at the midpoints of the sides.<\/li><li>The incircle is tangent to the nine-point circle of the triangle.<\/li><\/ul>\n\n\n\n<p>The inradius can be used to solve a variety of problems involving triangles, such as finding the area of the triangle, finding the length of a side of the triangle, or finding the distance between two points on a side of the triangle.<\/p>\n\n\n\n<p>Here is an example of how to use the inradius to find the area of a triangle:<\/p>\n\n\n\n<p>Given: A triangle with sides of length 3, 4, and 5.<\/p>\n\n\n\n<p>To find: The area of the triangle.<\/p>\n\n\n\n<p>Solution:<\/p>\n\n\n\n<p>1. Find the semiperimeter of the triangle: s = (3 + 4 + 5) \/ 2 = 6<\/p>\n\n\n\n<p>2. Find the area of the triangle: S = r * s = (6 * 3) \/ 2 = 9<\/p>\n\n\n\n<p>Therefore, the area of the triangle is 9.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><a href=\"https:\/\/www.practicemock.com\/ssc-cgl\/?ref=7334\"><img loading=\"lazy\" decoding=\"async\" width=\"3000\" height=\"375\" src=\"https:\/\/www.practicemock.com\/blog\/wp-content\/uploads\/2023\/05\/SSC-CGL-Free-Mock-2-1.png\" alt=\"\" class=\"wp-image-44253\"\/><\/a><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">Circumcenter of a Triangle<\/h2>\n\n\n\n<p>The circumcenter of a triangle is the center of the circle that passes through all three vertices of the triangle. The circumcenter is denoted by O.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.practicemock.com\/blog\/wp-content\/uploads\/2023\/05\/Circumcenter-of-a-triangle-1.png\" alt=\"\" class=\"wp-image-44212\" width=\"315\" height=\"314\"\/><\/figure>\n\n\n\n<p>The circumcenter can be found by intersecting the perpendicular bisectors of the sides of the triangle.<\/p>\n\n\n\n<p>The circumcenter has several properties, including:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>The circumcenter is equidistant from the three vertices of the triangle.<\/li><li>The circumradius is equal to the distance between the circumcenter and any vertex of the triangle.<\/li><li>The circumcircle is tangent to the three sides of the triangle at the midpoints of the sides.<\/li><\/ul>\n\n\n\n<p>The circumcenter can be used to solve a variety of problems involving triangles, such as finding the area of the triangle, finding the length of a side of the triangle, or finding the distance between two points on a side of the triangle.<\/p>\n\n\n\n<p>Here is an example of how to use the circumcenter to find the area of a triangle:<\/p>\n\n\n\n<p>Given: A triangle with vertices A, B, and C.<\/p>\n\n\n\n<p>To find: The area of the triangle.<\/p>\n\n\n\n<p>Solution:<\/p>\n\n\n\n<p>1. Find the circumcenter of the triangle: O<\/p>\n\n\n\n<p>2. Find the radius of the circumcircle: r = OA = OB = OC<\/p>\n\n\n\n<p>3. Find the area of the triangle: S = r * s = r * (AB + BC + CA) \/ 2<\/p>\n\n\n\n<p>Therefore, the area of the triangle is S.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><a href=\"https:\/\/www.practicemock.com\/ssc-cgl\/?ref=7334\"><img loading=\"lazy\" decoding=\"async\" width=\"3000\" height=\"375\" src=\"https:\/\/www.practicemock.com\/blog\/wp-content\/uploads\/2023\/05\/SSC-CGL-Free-Mock-2-1.png\" alt=\"\" class=\"wp-image-44253\"\/><\/a><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">Circumradius of a Triangle<\/h2>\n\n\n\n<p>The circumradius of a triangle is the radius of the circle that passes through all three vertices of the triangle. It is denoted by R.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.practicemock.com\/blog\/wp-content\/uploads\/2023\/05\/Circumradius-of-a-Triangle-1.png\" alt=\"\" class=\"wp-image-44213\" width=\"256\" height=\"255\"\/><\/figure>\n\n\n\n<p>The circumradius can be found using the following formula:<\/p>\n\n\n\n<p>R = 2S \/ (a + b + c)<\/p>\n\n\n\n<p>where S is the area of the triangle and a, b, and c are the lengths of the sides of the triangle.<\/p>\n\n\n\n<p>The circumradius has several properties, including:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>The circumradius is equal to the product of the semiperimeter of the triangle and the inradius.<\/li><li>The circumradius is equal to the distance between the circumcenter and any vertex of the triangle.<\/li><li>The circumcircle is tangent to the three sides of the triangle at the midpoints of the sides.<\/li><\/ul>\n\n\n\n<p>The circumradius can be used to solve a variety of problems involving triangles, such as finding the area of the triangle, finding the length of a side of the triangle, or finding the distance between two points on a side of the triangle.<\/p>\n\n\n\n<p>Here is an example of how to use the circumradius to find the area of a triangle:<\/p>\n\n\n\n<p>Given: A triangle with sides of length 3, 4, and 5.<\/p>\n\n\n\n<p>To find: The area of the triangle.<\/p>\n\n\n\n<p>Solution:<\/p>\n\n\n\n<p>1. Find the semiperimeter of the triangle: s = (3 + 4 + 5) \/ 2 = 6<\/p>\n\n\n\n<p>2. Find the inradius: r = S \/ s = (6 * 3) \/ 2 = 9<\/p>\n\n\n\n<p>3. Find the circumradius: R = 2S \/ (a + b + c) = 2 * 9 \/ (3 + 4 + 5) = 6<\/p>\n\n\n\n<p>4. Find the area of the triangle: S = r * R = 9 * 6 = 54<\/p>\n\n\n\n<p>Therefore, the area of the triangle is 54.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><a href=\"https:\/\/www.practicemock.com\/ssc-cgl\/?ref=7334\"><img loading=\"lazy\" decoding=\"async\" width=\"3000\" height=\"375\" src=\"https:\/\/www.practicemock.com\/blog\/wp-content\/uploads\/2023\/05\/SSC-CGL-Free-Mock-2-1.png\" alt=\"\" class=\"wp-image-44253\"\/><\/a><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">Orthocenter of a Triangle<\/h2>\n\n\n\n<p>The orthocenter of a triangle is the point of intersection of the altitudes of the triangle. An altitude is a line that is drawn from a vertex of a triangle and is perpendicular to the opposite side. Therefore, a triangle has three altitudes, one from each vertex. The orthocenter is denoted by H.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"250\" height=\"202\" src=\"https:\/\/www.practicemock.com\/blog\/wp-content\/uploads\/2023\/05\/Orthocenter-of-a-Triangle-1.png\" alt=\"\" class=\"wp-image-44214\"\/><\/figure>\n\n\n\n<p>The orthocenter has several properties, including:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>The orthocenter is always inside the triangle if the triangle is acute.<\/li><li>The orthocenter is always outside the triangle if the triangle is obtuse.<\/li><li>The orthocenter is at the vertex of the right angle if the triangle is right.<\/li><\/ul>\n\n\n\n<p>The orthocenter can be used to solve a variety of problems involving triangles, such as finding the area of the triangle, finding the length of a side of the triangle, or finding the distance between two points on a side of the triangle.<\/p>\n\n\n\n<p>Here is an example of how to use the orthocenter to find the area of a triangle:<\/p>\n\n\n\n<p>Given: A triangle with vertices A, B, and C.<\/p>\n\n\n\n<p>To find: The area of the triangle.<\/p>\n\n\n\n<p>Solution:<\/p>\n\n\n\n<p>1. Find the orthocenter of the triangle: H<\/p>\n\n\n\n<p>2. Find the altitude from vertex A to side BC: h<\/p>\n\n\n\n<p>3. Find the altitude from vertex B to side AC: k<\/p>\n\n\n\n<p>4. Find the area of the triangle: S = (1\/2) * h * k<\/p>\n\n\n\n<p>Therefore, the area of the triangle is S.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><a href=\"https:\/\/www.practicemock.com\/ssc-cgl\/?ref=7334\"><img loading=\"lazy\" decoding=\"async\" width=\"3000\" height=\"375\" src=\"https:\/\/www.practicemock.com\/blog\/wp-content\/uploads\/2023\/05\/SSC-CGL-Free-Mock-2-1.png\" alt=\"\" class=\"wp-image-44253\"\/><\/a><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">Questions for Practice<\/h2>\n\n\n\n<p>Here are some questions on Incircle, Inradius, Circumcenter, Circumradius &amp; Orthocenter.<\/p>\n\n\n\n<p>1. If \u2018I\u2019 the incentre of triangle ABC such that angle BAC = 44<sup>o<\/sup>, then find the measure of angle BIC.<\/p>\n\n\n\n<p>A) 112<sup>o<\/sup><\/p>\n\n\n\n<p>B) 68<sup>o<\/sup><\/p>\n\n\n\n<p>C) 88<sup>o<\/sup><\/p>\n\n\n\n<p>D) 44<sup>o<\/sup><\/p>\n\n\n\n<p>2. If the length of three sides of a triangle are 8 cm, 10 cm and 12 cm while the inradius of the triangle is 5 cm, then find the area of the triangle.<\/p>\n\n\n\n<p>&nbsp;A) 37.5 cm<sup>2<\/sup><\/p>\n\n\n\n<p>B) 150 cm<sup>2<\/sup><\/p>\n\n\n\n<p>C) 75 cm<sup>2<\/sup><\/p>\n\n\n\n<p>D) 300 cm<sup>2<\/sup><\/p>\n\n\n\n<p>3. If \u2018M\u2019 is the circum-centre of triangle PQR such that angle QPR = 36<sup>o<\/sup>, then find the measure of angle QMR.<\/p>\n\n\n\n<p>A) 18<sup>o<\/sup><\/p>\n\n\n\n<p>B) 72<sup>o<\/sup><\/p>\n\n\n\n<p>C) 108<sup>o<\/sup><\/p>\n\n\n\n<p>D) 60<sup>o<\/sup><\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><a href=\"https:\/\/www.practicemock.com\/ssc-cgl\/?ref=7334\"><img loading=\"lazy\" decoding=\"async\" width=\"3000\" height=\"375\" src=\"https:\/\/www.practicemock.com\/blog\/wp-content\/uploads\/2023\/05\/SSC-CGL-Free-Mock-2-1.png\" alt=\"\" class=\"wp-image-44253\"\/><\/a><\/figure>\n\n\n\n<p>4. If the length of three sides of a triangle is \u2018a\u2019 cm, 12 cm and 15 cm, respectively such that its area is 54 cm<sup>2<\/sup>, then find the value of given that the length of circum-radius of the triangle is 7.5 cm.<\/p>\n\n\n\n<p>A) 9 cm<\/p>\n\n\n\n<p>B) 4.5 cm<\/p>\n\n\n\n<p>C) 18 cm<\/p>\n\n\n\n<p>D) 10 cm<\/p>\n\n\n\n<p>5. If \u2018O\u2019 is the ortho-centre of triangle ABC, such that angle BOC = 50<sup>o<\/sup>, then find the measure of angle BAC.<\/p>\n\n\n\n<p>A) 120<sup>o<\/sup><\/p>\n\n\n\n<p>B) 100<sup>o<\/sup><\/p>\n\n\n\n<p>C) 110<sup>o<\/sup><\/p>\n\n\n\n<p>D) 130<sup>o<\/sup><\/p>\n\n\n\n<p>1. Ans: A<\/p>\n\n\n\n<p>Solution:<\/p>\n\n\n\n<p>Since, \u2018I\u2019 is the in-centre of the circle.<\/p>\n\n\n\n<p>So, angle BIC = 90<sup>o<\/sup> + (1\/2) \u00d7 angle BAC = 90<sup>o<\/sup> + (1\/2) \u00d7 44<sup>o<\/sup> = 90<sup>o<\/sup> + 22<sup>o<\/sup> = 112<sup>o<\/sup><\/p>\n\n\n\n<p>2. Ans: C<\/p>\n\n\n\n<p>Solution:<\/p>\n\n\n\n<p>Inradius of a triangle = {area of a triangle\/semi perimeter of the triangle}<\/p>\n\n\n\n<p>Since, semi-perimeter of the given triangle = (8 + 12 + 10)\/2 = 15 cm<\/p>\n\n\n\n<p>Area of the triangle = (inradius of the triangle) \u00d7 (semi-perimeter of the triangle) = 15 \u00d7 5 = 75 cm<sup>2<\/sup><\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><a href=\"https:\/\/www.practicemock.com\/ssc-cgl\/?ref=7334\"><img loading=\"lazy\" decoding=\"async\" width=\"3000\" height=\"375\" src=\"https:\/\/www.practicemock.com\/blog\/wp-content\/uploads\/2023\/05\/SSC-CGL-Free-Mock-2-1.png\" alt=\"\" class=\"wp-image-44253\"\/><\/a><\/figure>\n\n\n\n<p>3. Ans: B<\/p>\n\n\n\n<p>Solution:<\/p>\n\n\n\n<p>Since, \u2018M\u2019 is the circumradius of triangle PQR<\/p>\n\n\n\n<p>So, angle QMR = 2 \u00d7 (angle QPR) = 2 \u00d7 36<sup>o<\/sup> = 72<sup>o<\/sup><\/p>\n\n\n\n<p>4. Ans: A<\/p>\n\n\n\n<p>Solution:<\/p>\n\n\n\n<p>If length of three sides of a triangle are \u2018x\u2019 units, \u2018y\u2019 units and \u2018z\u2019 units, then<\/p>\n\n\n\n<p>Length of circumradius of a triangle = {(x \u00d7 y \u00d7 z) \u00f7 (4 \u00d7 area of the triangle)}<\/p>\n\n\n\n<p>So, (a \u00d7 12 \u00d7 15) \u00f7 (4 \u00d7 54) = 7.5<\/p>\n\n\n\n<p>So, a = 9<\/p>\n\n\n\n<p>5. Ans: D<\/p>\n\n\n\n<p>Solution:<\/p>\n\n\n\n<p>Since, \u2018O\u2019 is the ortho-centre of the triangle<\/p>\n\n\n\n<p>So, angle BOC = 180<sup>o<\/sup> \u2013 angle BAC<\/p>\n\n\n\n<p>So, 50<sup>o<\/sup> = 180<sup>o<\/sup> \u2013 angle BAC<\/p>\n\n\n\n<p>So, angle BAC = 180<sup>o<\/sup> \u2013 50<sup>o<\/sup> = 130<sup>o<\/sup><\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><a href=\"https:\/\/www.practicemock.com\/ssc-cgl\/?ref=7334\"><img loading=\"lazy\" decoding=\"async\" width=\"3000\" height=\"375\" src=\"https:\/\/www.practicemock.com\/blog\/wp-content\/uploads\/2023\/05\/SSC-CGL-Free-Mock-2-1.png\" alt=\"\" class=\"wp-image-44253\"\/><\/a><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>In SSC CGL exam, several questions are asked in Geometry. In this article, we will run you through several concepts related to Geometry along with examples so that you can understand them easily. Let\u2019s get started. Incircle of a triangle An incircle is a circle that is tangent to all three sides of a triangle. [&hellip;]<\/p>\n","protected":false},"author":3,"featured_media":44251,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"_uf_show_specific_survey":0,"_uf_disable_surveys":false,"footnotes":""},"categories":[24,26,16],"tags":[227],"class_list":["post-44208","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-exams","category-ssc-govt","category-ssc-cgl","tag-quantitative-aptitude"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v25.9 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>SSC CGL 2023- Important Quant Topics (Triangle) - Practicemock<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.practicemock.com\/blog\/ssc-cgl-2023-important-quant-topics-triangle\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"SSC CGL 2023- Important Quant Topics (Triangle) - Practicemock\" \/>\n<meta property=\"og:description\" content=\"In SSC CGL exam, several questions are asked in Geometry. In this article, we will run you through several concepts related to Geometry along with examples so that you can understand them easily. Let\u2019s get started. Incircle of a triangle An incircle is a circle that is tangent to all three sides of a triangle. 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In this article, we will run you through several concepts related to Geometry along with examples so that you can understand them easily. Let\u2019s get started. Incircle of a triangle An incircle is a circle that is tangent to all three sides of a triangle. 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In this article, we will run you through several concepts related to Geometry along with examples so that you can understand them easily. Let\u2019s get started. 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