{"id":623851,"date":"2023-08-31T11:34:16","date_gmt":"2023-08-31T11:34:16","guid":{"rendered":"https:\/\/www.indcareer.com\/schools\/?p=623851"},"modified":"2023-09-04T08:28:45","modified_gmt":"2023-09-04T08:28:45","slug":"kc-sinha-exercise-7-1-mathematics-solution-class-9-chapter-7-introduction-to-euclids-geometry","status":"publish","type":"post","link":"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-7-1-mathematics-solution-class-9-chapter-7-introduction-to-euclids-geometry\/","title":{"rendered":"KC Sinha: Exercise 7.1 – Mathematics Solution Class 9 Chapter 7 Introduction to Euclid’s Geometry"},"content":{"rendered":"\n

add <\/p>\n\n\n\n

Exercise 7.1<\/h3>\n\n\n\n

Type 1<\/p>\n\n\n\n

Question 1 <\/h4>\n\n\n\n

Which of the following statements are true and which are false ? Give reasons for your answer :<\/strong>
(i) These exists a point through which no line can pass.<\/strong>
Sol : False : Inf\u200cinite number of lines cam puss through a given point.<\/p>\n\n\n\n

(ii) A terminated line can be produced on either sides indef\u200cinitely.<\/strong>
Sol : True : By Euclid’s second postulate.<\/p>\n\n\n\n

(iii) With a given point A as centre, one and only one circle exists.<\/strong>
Sol : False : With the given point as centre and different radii , infinite number of circles can be drawn<\/p>\n\n\n\n

(iv) There exist two numbers x and y such that x \u2260 y but 2x = 2y.<\/strong>
Sol : False : Since 2x=2y \u21d2 x=y [Divide both sides by 2]<\/p>\n\n\n\n

(v) Two different lines, can have at most one point common.<\/strong>
Sol : True : By theorem 7.1<\/p>\n\n\n\n

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Question 2<\/h4>\n\n\n\n

What do you understand by Axioms and Postulates ?<\/strong>
Sol :
In Geometry, “Axiom<\/strong>” and “Postulate<\/strong>” are essentially interchangeable. In antiquity, they referred to propositions that were “obviously true” and only had to be stated, and not proven. In modern mathematics there is no longer an assumption that axioms are “obviously true”. Axioms are merely ‘background’ assumptions we make. The best analogy I know is that axioms are the “rules of the game”.<\/p>\n\n\n\n

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Question 3<\/h4>\n\n\n\n

What is a theorem ?<\/strong>
Sol:
A theorem<\/strong> is a logical consequence of the axioms. In Geometry, the “propositions” are all theorems: they are derived using the axioms and the valid rules. A “Corollary” is a theorem that is usually considered an “easy consequence” of another theorem. What is or is not a corollary is entirely subjective. Sometimes what an author thinks is a ‘corollary’ is deemed more important than the corresponding theorem. (The same goes for “Lemma<\/strong>“s, which are theorems that are considered auxiliary to proving some other, more important in the view of the author, theorem).<\/p>\n\n\n\n

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Question 4<\/h4>\n\n\n\n

Write some terms which are not def\u200cined in geometry.<\/strong>
Sol : Point , Line and Plane<\/p>\n\n\n\n

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Question 5<\/h4>\n\n\n\n

Write Euclid’s f\u200cive postulates.<\/strong>
Sol :
In Euclid’s Geometry, the main axioms\/postulates are:<\/p>\n\n\n\n

    \n
  1. Given any two distinct points, there is a line that contains them.<\/li>\n\n\n\n
  2. Any line segment can be extended to an infinite line.<\/li>\n\n\n\n
  3. Given a point and a radius, there is a circle with center in that point and that radius.<\/li>\n\n\n\n
  4. All right angles are equal to one another.<\/li>\n\n\n\n
  5. If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. (The parallel postulate<\/em>).<\/li>\n<\/ol>\n\n\n\n
    \n\n\n\n

    Question 6<\/h4>\n\n\n\n

    Write Euclid’s three axioms.<\/strong>
    Sol :<\/p>\n\n\n\n