5. Introduction to Euclid s Geometry

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1 This is only a sample of the book. For complete book contact us Introduction to Euclid s Geometry EXERISE 5.1 One Mark Questions 1. How many lines can pass through two distinct points? Sol. Only one line can pass through two distinct points.. How many lines can pass through a given point? Sol. Infinitely many lines can pass through a given point. 3. How many line segments can be determined by three collinear points? Sol. Three 4. How many line segments can be determined by three given non-collinear points? Sol. Three 5. Explain when a system of axioms is called consistent. (015-7SR3MM, 40ZL433) Sol. A system of axioms is called consistent, if one axiom does not contradict the other axioms of the system. 6. Name the term used by Euclid for boundary of the surface. (015-47TTMH) Sol. Term used is plane. Two Marks Questions 7. Prove that an equilateral triangle can be constructed on any given line segment. (015-EFK16AA; 014-1KMUD4) Or Apply Euclid s postulates to prove that an equilateral triangle can be constructed on any given line segment. (014-H7V08T) Sol. Let a line segment AB be given. Here, we need to prove that an equilateral triangle can be constructed. A B A B A B (i) (ii) (iii) Proof : With A as centre and AB as radius, we can construct a circle, using Euclid s postulate 3. Hence, we draw a circle with A as centre and AB as radius. Similarly, using Euclid s postulate 3, we draw a circle with B as centre and BA as radius. Let the two circles (in fact arcs) intersect at a point say. Now, we join A and B and obtain the triangle AB. Now, we have A = AB (Radii of the same circle)...(1) and BA = B (Radii of the same circle)...() (1)

2 So, from (1) and (), AB = B = A (Using Euclid s axiom that things which are equal to the same thing are equal) Hence, AB is an equilateral triangle. 8. P and Q are centres of the two intersecting circles which intersect at R (see figure). Prove that PQ = QR = PR. R (015-7RRM1WO, TYYZTX8; 014-1Z4HTB3) P Q Sol. We have : PQ = PR ENTERPRISES (Radii of the same circle with centre P)...(1) and PQ = QR (Radii of the same circle with centre Q)...() So, from (1) and (), PQ = QR = PR (Things which are equal to the same thing are equal) 9. If P, Q and R are three points on a line and Q lies between P and R, then prove that PQ + QR = PR (see figure). P Q R Sol. In the given figure, PR coincides with PQ + QR. But we know from Euclid s axiom (4) that things which coincide with one another are equal to one another. So, we can conclude that PQ + QR = PR. Note : In this solution, it has been assumed that there is a unique line passing through two given points. 10. Define the terms and also draw them : (i) Parallel lines, (ii) Perpendicular lines. (015-VLDANHG; 014-JTJUKRX) Sol. (i) Parallel lines : Lines in a plane which do not intersect are called parallel lines. In the figure, l and m are parallel lines. ENTERPRISES (015-99HU3ZJ, T6VL9NN; 014-IVQY5PP) m (ii) Perpendicular lines : Two lines are said to be perpendicular, if the angle between them is of 90. In the figure, AB and D are perpendicular lines. A l B D 11. Ram and Ravi have the same weight. If they each gain weight by kg, how will their new weights be compared? State Euclid s axiom used. (015-3O09O4) Sol. New weights will be the same. Euclid axiom : When equals are added to equals the totals (sums) are equal.

3 1. State Euclid s fifth postulate. (015-8QPWJP1, BUMHEM, HBHA; 01-4) Sol. Euclid s fifth postulate : If a straight line l falls on two straight lines m and n such that the sum of interior angles on the same side of l is less than 180, then the lines if produced, will meet on this side of l. 13. onsider two postulates given below : (i) Given any two distinct points R and S, there exists a third point T which is in between R and S. (ii) There exist at least three points which are not in the same straight line. Now, answer the following questions : (a) Do these postulates contain any undefined terms? (b) Do they follow from Euclid s postulates? Explain. (015-OTT6UG, J1Q09O; 014 EKWGZJL, ZKOGS8A; UNA6S, OQAFMAK) Sol. (a) Point and straight line are ENTERPRISES undefined terms. (b) These postulates do not follow from Euclid s postulates. However, they follow from the axiom which states that Given two distinct points, there is a unique line that passes through them. 14. Prove that Two distinct lines cannot have more than one point in common. (015-EUJPWWX) Sol. Let us assume that the two lines l and m have more than one point, i.e., say two points P and Q in common. Then, l is a line passing through two points P and Q. Also, we have a line m which is a line passing through P and Q. Thus, we get two distinct lines l and m passing through two given points P and Q, which is a contradiction to the fact that through two given points, there passes one and only one line. Hence, our assumption is wrong. Hence, the result. 15. State any two Euclid s axioms. (015-NFR5B7H) Sol. Euclid s axioms : (1) Whole is greater than the part. () When equals are subtracted from equals, the remainders are equal. 16. In the figure, AB = D. Prove that A = BD. Also, write the used Euclid s axiom. ENTERPRISES (015-PZN9YJB, GOVDEM) A B D Sol. AB = D So, AB B = D B A = BD Euclid s axioms used : When equals are subtracted from equals, remainders are equal. 17. In the figure, A = XD, is the mid-point of AB A X and D is the mid-point of XY. Using an Euclid s axiom, show that AB = XY. (014 L5L1EU8; 013 TZWM1AM, MZE3UO, BYZU8F1, D 3VUNKU7; 01 39, 53, 67; , 15; A1) Sol. AB = A ( is the mid-point of AB) B Y XY = XD (D is the mid-point of XY) Also, A = XD (Given) Therefore, AB = XY (Things which are double of the same thing are equal to one another)

4 18. Define the term POLYGON and ANGLE. (015-YQU8ZQQ) Sol. Polygon : A simple closed figure made of line segments only. Angle : A figure formed by two rays with a common initial point. 19. In the figure, if OX = 1 XY, PX = 1 XZ and OX = PX, then using Euclid s axioms, show that XY = XZ. X Y O Sol. OX = 1 XY... (1) P Z ENTERPRISES (015-3HW1TZ, JNUUHUG; 014 IL573QT; 013 TZWM1AM) PX = 1 XZ... () OX = PX... (3) From (1), () and (3), we have : 1 XY = 1 XZ (Things which are equal to the same (or equal) thing are equal to one another) Therefore, XY = XZ (Things which are double of the same thing are equal to one another) 0. If a point lies between two points A and B such that A = B, then prove that A = 1 AB. Explain by drawing the figure. ( BXZ, GIQQUB; 014 GGRO93; 013 SFBLY, BFMFPM7, 8HGGD8, JFA76T4; 01 0, 36; , 0, 6, 33; A1, A, 14-B1) Sol. Given A = B. AENTERPRISES B So, A + A = B + A (If equals are added to equals the wholes are equals) or A = AB Hence, A = 1 AB 1. Prove that every line segment has one and only one mid-point. (015 IRUZPDV; 014 5JY7AP3; 013 X9PA10E; 01 50; ; ) Sol. Suppose and are two mid-points of a line segment AB. Then, A = 1 AB A B and A = 1 AB

5 A = A' [Things which are equal to the same thing are equal to one another.] This is possible only when and coincide. Hence, every line segment has one and only one mid-point.. In the given figure, if point lies between A and B, then prove that AB > A. Which Euclid s axiom is applied? A B (015-6KBELZ0, QJ5TFWF; 014 DPEZBK; 01 63) Sol. In the given figure, lies between A and B. If AB is a whole, A is a part of AB. According to Euclid s axiom (5), which states that the whole is greater than the part, AB > A. 3. Solve the equation x 15 = 5 ENTERPRISES and state Euclid s axiom used here. (015-FA01BZ; 014 IL7R8K9, AVHHDG9; 013 VIPDEL; 01 69) Sol. x 15 = 5 According to Euclid s axiom (), if equals are added to equals, wholes are equal. So, adding 15 to both the sides, we get x = or x =40 Three Marks Question 4. (a) State Euclid s fifth postulate. (b) Solve u 5 = 15 and state the axiom used here. Sol. (a) If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the sum of the angles is less than two right angles. (b) u 5 = 15 So, u = (Euclid s axiom : If equals are added to equals, the wholes are equal) u + 0 = 0 ENTERPRISES u = 0, which is the required solution. Four Marks Questions 5. It is known that a c = 30 and that a = b. Show that b c = 30. Write the Euclid s axiom that best illustrates this statement. Also, give two more axioms other than the axiom used in the above situation. (015-GJMA9X, 7ZXU1TQ) Sol. As a = b, so a c = b c But a c =30 Therefore, b c =30 Euclid s axiom : Things which are equal to the same thing are equal. Other two axioms : Same as in solution of Q. 15 on Page Two salesmen make equal sales during the month of June. In July, each salesmen doubles his sale of the month of June. ompare their sales in July. State which axiom you use here. Also, give two more axioms other than the axiom used in the above situation. (015-7SR3MM, WXD7ZVQ, JNUUHUG)

6 Sol. Their sales in July are equal. Euclid s axiom used : Double of the same thing are equal. Other two axioms : Same as in solution of Q. 15 on Page It is known that if a + b = 10 then a + b c = 10 c. Write the Euclid s axiom that best illustrates this statement. Also, give two more axioms other than the axiom used in the above situation. (015-61OPML, 76HMM3X, 6KBELZO) Sol. Euclid s axiom : When equal are subtracted (here c), the remainders are equal. Other two axioms : (1) Double of the same thing are equal. () The whole is greater than the part. 8. A square is a POLYGON made up of four LINE SEGMENTS, out of which, length of three line segments are equal to the length of fourth one and all its ANGLES are RIGHT ANGLES. ENTERPRISES Define the terms used in this definition which have been highlighted. (015-HBHA) Sol. Polygon : A simple closed figure made of line segments only. Line segment : A part of a line having two end-points. Angle : A figure formed by two rays with a common initial point. Right angle : An angle of measure 90. PRATIE EXERISE One Mark Question 1. The things which are double of same thing are : (a) equal (b) halves of same thing (c) unequal (d) double of the same thing Two Marks Questions. Solve the equation x + 4 = 10 and state Euclid s axiom used. (015-USDH3UW) Four Marks Questions 8. Sunil and Shyam have the same ENTERPRISES weight. If they each gain weight by 5 kg, how will their new weights be compared using an axiom? Write the Euclid s axiom that best supports your answer. Also, give two more axioms other than the axiom used in the above situation. (015-40ZL433, AMIHEM) 11. It is known that if a + c = 5 and that c = b. Show that a + b = 5. Write the Euclid s axiom that best illustrates this statement. Also, give two more axioms other than the axiom used in the above situation. (015-T57UAD, QHYQUU) 1. Using Euclid s axiom, compare lengths AD and AF. State which axiom you use here. Also, give two more axioms other than the axiom used in the above situation. A B D E F G H (015-AQPK45K) Ans. 1. Equal. x = 6; if equals are subtracted from equals, the remainders are equal. 3. If equals are added, the totals are equal. 4. If equals are added, the totals are equal.

7 VALUE BASED QUESTIONS 5. Teacher held two sticks AB and D of equal length in her hands and marked their mid-points M and N respectively. She then asked the students whether AM is equal to ND or not. Arpita answered yes. Is Arpita correct? State axiom of Euclid s that support her answer. Which characteristics of Arpita you want to inculcate in your nature? (015-8GDMGWT) Sol. Yes. Euclid s axiom : Halves of equals are equal. Value or characteristics : Quick reply with confidence. 6. Rohan s maid has two children. Both of them have equal number of dresses. Rohan on his birthday plans to give both of them same number of dresses. What you can say about the number of dresses ENTERPRISES each of them will have after Rohan s birthday? Which Euclid axiom is used to answer this question? What value is Rohan depicting by doing so? Write two more Euclid axioms. (015-AMIHEM) Sol. They are equal. Euclid s axiom : When equals are added, the totals are equal. Value : Kind towards poor workers. Other two axioms : Same as in solution of Q. 15 of Exercise Varun and Lokesh contributed equal amount towards Prime Minister Relief Fund. Lokesh and Anmol contributed equal amount towards Prime Minister Relief Fund. If Anmol conributed ` 1500, how much Varun contributed? What value they all are exhibiting by doing so? Which Euclid s axiom did help you in reaching the correct answer? State the exact axiom. State third postulate of Euclid. Sol. Varun contribution is ` Value : Awareness to help the needy people in national calamity. Axiom : Things equal to the same thing are equal. Third postulate : A circle can be drawn with any centre and any radius. ENTERPRISES Q.. Match the following : olumn I olumn II 1. Undefined terms A. Postulates. Father of Geometry B. Theorems 3. Assumptions specified to Geometry. Surface 4. ommon notions D. Point, line, plane 5. Having length and breadth only E. Euclid 6. Edges of a surface F. Axioms 7. Euclid s famous treatise G. Elements 8. Statements having proof H. Lines Sol. 1. D;. E; 3. A; 4. F; 5. ; 6. H; 7. G; 8. B. ontd...

8 Practice Paper 1 (Solved) MATHEMATIS Time Allowed : 3 hours LASS IX Maximum Marks : 90 General Instructions : (i) All questions are compulsory. (ii) The question paper consists of 31 questions divided into four sections A, B, and D. Section A comprises of 4 questions of 1 mark each; Section B comprises of 6 questions of marks each; Section comprises of 10 questions of 3 marks each and Section D comprises of 11 questions of 4 marks each. (iii) There is no overall choice in this question ENTERPRISES paper. (iv) Use of calculator is not permitted. SETION A Question numbers 1 to 4 carry one mark each. 1. Find a rational number between and 3. Sol. A rational number between and 3 is + 3, i.e., 5.. Which of the following is a polynomial? ( a) p 3/ 3 3 3x ( b) 3x 1 ( c) x + 3 p x ( d) y + 3y Sol. (c) x x 3 / x 3. What is the distance of the point (0, 3) from the origin? Sol. Distance of the point (0, 3) from the origin (0, 0) = 3 units. SETION B Question numbers 5 to 10 carry ENTERPRISES two marks each. 4. Represent on the number line. Sol. Draw a right triangle OAB such that OA = 1 unit, AB = 1 unit and AB OA. Then, join OB. With O as centre and OB as radius, draw an arc to intersect the number line at. Then, represents. 5. In a triangle PQR, QO and RO are the bisectors of PQR and PRQ, such that RQO = QRO and PQO = PRO. Show that : PQR = PRQ Sol. RQO = QRO...(1) (Given) PQO = PRO...() (Given) Adding (1) and (), we get RQO + PQO = QRO + PRO PQR = PRQ Hence, proved.

5. Introduction to Euclid s Geometry

5. Introduction to Euclid s Geometry Multiple Choice Questions CBSE TREND SETTER PAPER _ 0 EXERCISE 5.. If the point P lies in between M and N and C is mid-point of MP, then : (A) MC + PN = MN (B) MP +