{"id":4297,"date":"2023-02-07T09:22:56","date_gmt":"2023-02-07T09:22:56","guid":{"rendered":"https:\/\/theexampillar.com\/ncert\/?p=4297"},"modified":"2023-02-07T09:23:42","modified_gmt":"2023-02-07T09:23:42","slug":"ncert-solutions-class-9-maths-chapter-5-introduction-to-euclids-geometry-ex-5-1","status":"publish","type":"post","link":"https:\/\/theexampillar.com\/ncert\/ncert-solutions-class-9-maths-chapter-5-introduction-to-euclids-geometry-ex-5-1\/","title":{"rendered":"NCERT Solutions Class 9 Maths Chapter 5 Introduction to Euclids Geometry Ex 5.1"},"content":{"rendered":"<h2 style=\"text-align: center;\"><span style=\"color: #000000; font-family: georgia, palatino, serif;\">NCERT Solutions Class 9 Maths\u00a0<\/span><br \/>\n<span style=\"color: #000000; font-family: georgia, palatino, serif;\">Chapter &#8211; 5 (Introduction to Euclids Geometry)\u00a0<\/span><\/h2>\n<p style=\"text-align: justify;\"><span style=\"color: #000000; font-family: georgia, palatino, serif;\">The NCERT Solutions in English Language for Class 9 Mathematics\u00a0<strong>Chapter &#8211; 5 Introduction to Euclids Geometry <\/strong>Exercise 5.1 has been provided here to help the students in solving the questions from this exercise.\u00a0<\/span><\/p>\n<p><span style=\"color: #000000;\"><strong>Chapter 5: Introduction to Euclid Geometry <\/strong><\/span><\/p>\n<ul>\n<li><a href=\"https:\/\/theexampillar.com\/ncert\/ncert-solutions-class-9-maths-chapter-5-introduction-to-euclids-geometry-ex-5-2\/\" target=\"_blank\" rel=\"noopener\">NCERT Solution Class 9 Maths Ex &#8211; 5.2<\/a><\/li>\n<\/ul>\n<h2 style=\"text-align: center;\"><span style=\"color: #000000; font-family: georgia, palatino, serif;\">Exercise &#8211; 5.1\u00a0<\/span><\/h2>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\"><strong>1. Which of the following statements are true and which are false? Give reasons for your answers.<br \/>\n<\/strong><strong>(i) <\/strong>Only one line can pass through a single point.<br \/>\n<strong>(ii)<\/strong> There are an infinite number of lines which pass through two distinct points.<br \/>\n<strong>(iii)<\/strong> A terminated line can be produced indefinitely on both the sides.<br \/>\n<strong>(iv)<\/strong> If two circles are equal, then their radii are equal.<br \/>\n<strong>(v)<\/strong> In Fig. 5.9, if AB = PQ and PQ = XY, then AB = XY.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-4302\" src=\"https:\/\/theexampillar.com\/ncert\/wp-content\/uploads\/2023\/01\/NCERT-Solutions-Class-9-Maths-Ex-5.1-Q1.png\" alt=\"NCERT Class 9 Solutions Maths\" width=\"719\" height=\"169\" srcset=\"https:\/\/theexampillar.com\/ncert\/wp-content\/uploads\/2023\/01\/NCERT-Solutions-Class-9-Maths-Ex-5.1-Q1.png 719w, https:\/\/theexampillar.com\/ncert\/wp-content\/uploads\/2023\/01\/NCERT-Solutions-Class-9-Maths-Ex-5.1-Q1-300x71.png 300w, https:\/\/theexampillar.com\/ncert\/wp-content\/uploads\/2023\/01\/NCERT-Solutions-Class-9-Maths-Ex-5.1-Q1-480x113.png 480w\" sizes=\"auto, (max-width: 719px) 100vw, 719px\" \/><br \/>\n<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\"><strong>Answer &#8211;<\/strong> <\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\"><strong>(i) <\/strong>Only one line can pass through a single point.<br \/>\n<strong>(False) <\/strong>There can be infinite number of lines that can be drawn through a single point. Hence, the statement mentioned is False<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\"><strong>(ii)<\/strong> There are an infinite number of lines which pass through two distinct points.<br \/>\n<strong>(False)\u00a0<\/strong>Through two distinct points, there can be only one line that can be drawn. Hence, the statement mentioned is False<br \/>\n<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\"><strong>(iii)<\/strong> A terminated line can be produced indefinitely on both the sides.<br \/>\n<strong>(True)\u00a0<\/strong>A line that is terminated can be indefinitely produced on both sides as a line can be extended on both its sides infinitely. Hence, the statement mentioned is True.<br \/>\n<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\"><strong>(iv)<\/strong> If two circles are equal, then their radii are equal.<br \/>\n<strong>(True)\u00a0<\/strong>The radii of two circles are equal when the two circles are equal. The circumference and the centre of both the circles coincide; and thus, the radius of the two circles should be equal. Hence, the statement mentioned is True.<br \/>\n<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\"><strong>(v)<\/strong> In Fig. 5.9, if AB = PQ and PQ = XY, then AB = XY.<br \/>\n<strong>(True) <\/strong>According to Euclid\u2019s 1<sup>st<\/sup>\u00a0axiom- \u201cThings which are equal to the same thing are also equal to one another\u201d. Hence, the statement mentioned is True.<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\"><strong>2. Give a definition for each of the following terms. Are there other terms that need to be defined first? What are they, and how might you define them?<br \/>\n<\/strong><strong>(i) parallel lines<br \/>\n<\/strong><strong>(ii) perpendicular lines<br \/>\n<\/strong><strong>(iii) line segment<br \/>\n<\/strong><strong>(iv) radius of a circle<br \/>\n<\/strong><strong>(v) square<\/strong><\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\"><strong>Answer &#8211;<\/strong> \u00a0We have to define \u2018Ray\u2019, \u2018Straight line\u2019 and a \u2018point\u2019.<br \/>\n<strong>Ray:<\/strong> A part of a line, which starts at a point (Here A) and goes off in a particular direction to infinity possibly through a second point (B in this case).<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-4303 aligncenter\" src=\"https:\/\/theexampillar.com\/ncert\/wp-content\/uploads\/2023\/01\/NCERT-Solutions-Class-9-Maths-Ex-5.1-Q2a.png\" alt=\"NCERT Class 9 Solutions Maths\" width=\"249\" height=\"76\" \/><\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\"><strong>Straight Line:<\/strong> The basic concept about a line is that it should be straight, and it can be extended infinitely in both directions.<img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-4304 aligncenter\" src=\"https:\/\/theexampillar.com\/ncert\/wp-content\/uploads\/2023\/01\/NCERT-Solutions-Class-9-Maths-Ex-5.1-Q2b.png\" alt=\"NCERT Class 9 Solutions Maths\" width=\"240\" height=\"82\" \/><\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\"><strong>Point:<\/strong> A small dot made by a sharp pencil on a sheet of paper gives an idea about a point.<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\"><strong>(i) Parallel lines \u2013<\/strong> Parallel lines are those lines which never intersect each other and are always at a constant perpendicular distance between each other. Parallel lines can be two or more lines.<img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-4305 aligncenter\" src=\"https:\/\/theexampillar.com\/ncert\/wp-content\/uploads\/2023\/01\/NCERT-Solutions-Class-9-Maths-Ex-5.1-Q2c.png\" alt=\"NCERT Class 9 Solutions Maths\" width=\"321\" height=\"246\" srcset=\"https:\/\/theexampillar.com\/ncert\/wp-content\/uploads\/2023\/01\/NCERT-Solutions-Class-9-Maths-Ex-5.1-Q2c.png 321w, https:\/\/theexampillar.com\/ncert\/wp-content\/uploads\/2023\/01\/NCERT-Solutions-Class-9-Maths-Ex-5.1-Q2c-300x230.png 300w\" sizes=\"auto, (max-width: 321px) 100vw, 321px\" \/><strong>(ii) Perpendicular lines \u2013<\/strong> Perpendicular lines are those lines which intersect each other in a plane at right angles. The lines are said to be perpendicular to each other.<img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-4306 aligncenter\" src=\"https:\/\/theexampillar.com\/ncert\/wp-content\/uploads\/2023\/01\/NCERT-Solutions-Class-9-Maths-Ex-5.1-Q2d.png\" alt=\"NCERT Class 9 Solutions Maths\" width=\"270\" height=\"226\" \/><strong>(iii) Line segment \u2013<\/strong> When a line cannot be extended any further because of its two end points, then the line is known as a line segment. A line segment has 2 end points.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-4307 aligncenter\" src=\"https:\/\/theexampillar.com\/ncert\/wp-content\/uploads\/2023\/01\/NCERT-Solutions-Class-9-Maths-Ex-5.1-Q2e.png\" alt=\"NCERT Class 9 Solutions Maths\" width=\"266\" height=\"77\" \/><br \/>\n<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\"><strong>(iv) Radius of circle \u2013<\/strong> A radius of a circle is the line from any point on the circumference of the circle to the center of the circle.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-4308 aligncenter\" src=\"https:\/\/theexampillar.com\/ncert\/wp-content\/uploads\/2023\/01\/NCERT-Solutions-Class-9-Maths-Ex-5.1-Q2f.png\" alt=\"NCERT Class 9 Solutions Maths\" width=\"164\" height=\"161\" \/><strong>(v) Square \u2013<\/strong> A quadrilateral in which all the four sides are said to be equal, and each of its internal angles is a right angle, is called square.<img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-4309 aligncenter\" src=\"https:\/\/theexampillar.com\/ncert\/wp-content\/uploads\/2023\/01\/NCERT-Solutions-Class-9-Maths-Ex-5.1-Q2g.png\" alt=\"NCERT Class 9 Solutions Maths\" width=\"207\" height=\"225\" \/><strong>3. Consider two \u2018postulates\u2019 given below:<br \/>\n<\/strong><strong>(i)<\/strong> Given any two distinct points A and B, there exists a third point C which is in between A and B.<br \/>\n<strong>(ii)<\/strong> There exist at least three points that are not on the same line.<br \/>\n<strong>Do these postulates contain any undefined terms? Are these postulates consistent? Do they follow from Euclid\u2019s postulates? Explain.<\/strong><\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\"><strong>Answer &#8211;<\/strong> There are several undefined terms which we should keep in mind. They are consistent, because they deal with two different situations:<br \/>\n<strong>(i)<\/strong> says that the given two points A and B, there is a point C lying on the line in between them;<br \/>\n<strong>(ii)<\/strong> says that given A and B, we ca take C not lying on the line through A and B. These \u2018postulates\u2019 do not follow from Euclid\u2019s postulates. However, they follow from axiom stated as given two distinct points; there is a unique line that passes through them.<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\"><strong>4. If a point C lies between two points A and B such that AC = BC, then prove that AC = \u00bd AB. Explain by drawing the figure.<br \/>\n<\/strong><strong>Answer &#8211;<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-4310\" src=\"https:\/\/theexampillar.com\/ncert\/wp-content\/uploads\/2023\/01\/NCERT-Solutions-Class-9-Maths-Ex-5.1-Q4.png\" alt=\"NCERT Class 9 Solutions Maths\" width=\"434\" height=\"62\" srcset=\"https:\/\/theexampillar.com\/ncert\/wp-content\/uploads\/2023\/01\/NCERT-Solutions-Class-9-Maths-Ex-5.1-Q4.png 434w, https:\/\/theexampillar.com\/ncert\/wp-content\/uploads\/2023\/01\/NCERT-Solutions-Class-9-Maths-Ex-5.1-Q4-300x43.png 300w\" sizes=\"auto, (max-width: 434px) 100vw, 434px\" \/><br \/>\n<\/strong>Given that, AC = BC<br \/>\nNow, adding AC both sides.<br \/>\nL.H.S + AC = R.H.S + AC<br \/>\nAC + AC = BC+ AC<br \/>\n2AC = BC+AC<br \/>\nWe know that, BC+AC = AB (as it coincides with line segment AB)<br \/>\n\u2234 2 AC = AB (If equals are added to equals, the wholes are equal.)<br \/>\n\u21d2 AC = (\u00bd)AB.<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\"><strong>5. In Question 4, point C is called a mid-point of line segment AB. Prove that every line segment has one and only one mid-point.<br \/>\n<\/strong><strong>Answer &#8211;<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-4311\" src=\"https:\/\/theexampillar.com\/ncert\/wp-content\/uploads\/2023\/01\/NCERT-Solutions-Class-9-Maths-Ex-5.1-Q5.png\" alt=\"NCERT Class 9 Solutions Maths\" width=\"386\" height=\"48\" srcset=\"https:\/\/theexampillar.com\/ncert\/wp-content\/uploads\/2023\/01\/NCERT-Solutions-Class-9-Maths-Ex-5.1-Q5.png 386w, https:\/\/theexampillar.com\/ncert\/wp-content\/uploads\/2023\/01\/NCERT-Solutions-Class-9-Maths-Ex-5.1-Q5-300x37.png 300w\" sizes=\"auto, (max-width: 386px) 100vw, 386px\" \/><br \/>\n<\/strong>Let, AB be the line segment<br \/>\nAssume that points P and Q are the two different mid points of AB.<br \/>\nNow,<br \/>\n\u2234 P and Q are midpoints of AB.<br \/>\nTherefore,<br \/>\nAP = PB and AQ = QB.<br \/>\nalso,<br \/>\nPB + AP = AB (as it coincides with line segment AB)<br \/>\nSimilarly, QB+AQ = AB.<br \/>\nNow,<br \/>\nAdding AP to the L.H.S and R.H.S of the equation AP = PB<br \/>\nWe get, AP+AP = PB+AP (If equals are added to equals, the wholes are equal.)<br \/>\n\u21d2 2AP = AB\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 &#8212;&#8212;\u2014 (i)<br \/>\nSimilarly,<br \/>\n2 AQ = AB\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 &#8212;&#8212;\u2014 (ii)<br \/>\nFrom (i) and (ii), Since R.H.S are same, we equate the L.H.S<br \/>\n2 AP = 2 AQ (Things which are equal to the same thing are equal to one another.)<br \/>\n\u21d2 AP = AQ (Things which are double of the same things are equal to one another.)<br \/>\nThus, we conclude that P and Q are the same points.<br \/>\nThis contradicts our assumption that P and Q are two different mid points of AB.<br \/>\nThus, it is proved that every line segment has one and only one mid-point.<br \/>\nHence Proved.<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\"><strong>6. In Fig. 5.10, if AC = BD, then prove that AB = CD.<br \/>\n<\/strong><strong>Answer &#8211;<\/strong><\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-4312\" src=\"https:\/\/theexampillar.com\/ncert\/wp-content\/uploads\/2023\/01\/NCERT-Solutions-Class-9-Maths-Ex-5.1-Q6.jpeg\" alt=\"NCERT Class 9 Solutions Maths\" width=\"271\" height=\"104\" \/><br \/>\nIt is given, AC = BD<br \/>\nFrom the given figure, we get,<br \/>\nAC = AB+BC<br \/>\nBD = BC+CD<br \/>\n\u21d2 AB+BC = BC+CD [AC = BD, given]<br \/>\nWe know that, according to Euclid\u2019s axiom, when equals are subtracted from equals, remainders are also equal.<br \/>\nSubtracting BC from the L.H.S and R.H.S of the equation AB+BC = BC+CD, we get,<br \/>\nAB + BC &#8211; BC = BC + CD &#8211; BC<br \/>\nAB = CD<br \/>\nHence Proved.<\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\"><strong>7. Why is Axiom 5, in the list of Euclid\u2019s axioms, considered a \u2018universal truth\u2019? (Note that the question is not about the fifth postulate.)<\/strong><br \/>\n<strong>Answer &#8211; <\/strong>Axiom 5 of Euclid&#8217;s Axioms states that &#8211; \u201cThe whole is greater than the part.\u201d This axiom is known as a universal truth because it holds true in any field of mathematics and in other disciplinarians of science as well. AB is a whole part.<\/span><\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>NCERT Solutions Class 9 Maths\u00a0 Chapter &#8211; 5 (Introduction to Euclids Geometry)\u00a0 The NCERT Solutions in English Language for Class 9 Mathematics\u00a0Chapter &#8211; 5 Introduction to Euclids Geometry Exercise 5.1 has been provided here to help the students in solving the questions from this exercise.\u00a0 Chapter 5: Introduction to Euclid Geometry NCERT Solution Class 9 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[698],"tags":[709,702,186,710,701,5],"class_list":["post-4297","post","type-post","status-publish","format-standard","hentry","category-class-9-maths","tag-class-9th-chapter-5-introduction-to-euclids-geometry-in-english","tag-ncert-solution-class-9","tag-ncert-solutions","tag-ncert-solutions-class-9-maths-chapter-5-in-english","tag-ncert-solutions-class-9-maths-in-english","tag-ncert-solutions-in-english"],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v22.9 (Yoast SEO v27.4) - 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