Assignments in Mathematics Class IX (Term 2) 8. QUADRILATERALS

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1 Assignments in Mathematics Cass IX (Term 2) 8. QUADRILATERALS IMPORTANT TERMS, DEFINITIONS AND RESULTS Sum of the anges of a quadriatera is 360. A diagona of a paraeogram divides it into two congruent trianges. In a paraeogram, (i) opposite sides are equa (ii) opposite anges are equa (iii) diagonas bisect each other A quadriatera is a paraeogram, if (i) opposite sides are equa or (ii) opposite anges are equa or (iii) diagonas bisect each other or (iv) a pair of opposite sides is equa and parae Diagonas of a rectange bisect each other and are equa and vice-versa. Diagonas of a rhombus bisect each other at right anges and vice-versa. Diagonas of a square bisect each other at right anges and are equa, and vice-versa. The ine segment joining the mid-points of any two sides of a triange is parae to the third side and is haf of it. A ine through the mid-point of a side of a triange parae to another side bisects the third side. The quadriatera formed by joining the mid-points of the sides of a quadriatera, in order, is a paraeogram. SUMMATIVE ASSESSMENT MULTIPLE CHOICE QUESTIONS [1 Mark] 1. Two consecutive anges of a paraeogram are in the ratio 1 : 3. Then the smaer ange is : (a) 50 (b) 90 (c) 60 (d) A quadriatera is a paraeograms if : (a) both pairs of opposite sides are equa (b) both pairs of opposite anges are equa (c) the diagonas bisect each other (d) a of these 3. ABCD is a rhombus. Diagona AC is equa to one of its sides. Then ABC must be : (a) a right anged triange (b) an equiatera triange (c) an isoscees triange (d) none of these 4. In the figure, ABCD is a paraeogram. The vaues of x and y are respectivey : A. Important Questions (a) 80 (b) 90 (c) 50 (d) The sum of three anges of a quadriatera is 3 right anges. Then the fourth ange is a/an : (a) right ange (b) obtuse ange (c) acute ange (d) refex ange 7. In the given figure, the measure of DOC is equa to : (a) 70, 110 (b) 70, 70 (c) 110, 70 (d) 70, In the given figure, ABCD is a rhombus. If A = 80, then CDB is equa to : (a) 90 (b) 180 (c) 118 (d) Two adjacent anges of a paraeogram are 2x + 30 and 3x 30. Then the vaue of x is : (a) 30 (b) 60 (c) 0 (d) In the figure, AB DC, then the measure of D is equa to : 1

2 17. The sides BA and DC of a quadriatera ABCD are produced as shown in the figure. Then which of the foowing reations is true? (a) 70 (b) 140 (c) 180 (d) In the rectange ABCD, BAC = 4x, if BCA = 5x, then measures of ACD and CAD are respectivey : (a) 50, 40 (b) 40, 50 (c) 80, 100 (d) none of these 11. In the figure, D, E and F are the mid-points of the sides AB, BC and CA respectivey. If AC = 8.2 cm, then vaue of DE is : (a) x + y = (b) x y = (c) x + y = 2 ( ) (d) none of these 18. In the figure, P and Q are mid-points of sides AB and AC respectivey of ABC. If PQ = 3.5 cm and AB = AC = 9 cm, then the perimeter of ABC is : ASHA (a) 20 cm (b) 23 cm (c) 25 cm (d) 27 cm 19. In the figure, if ABCD is a square, then vaue of x is : (a) 8.2 cm (c) 2.05 cm (b) 4.1 cm (d) none of these 12. In a rectange ABCD, diagonas AC and BD intersect at O. If AO = 3 cm, then the ength of the diagona BD is equa to : (a) 3 cm (b) 9 cm (c) 6 cm (d) 12 cm 13. Three anges of a quadriatera are 75, 90 and 75.The fourth ange is : (a) 90 (b) 95 (c) 105 (d) Diagonas of a paraeogram ABCD intersect at O. If BOC = 90 and BDC = 50, then OAB is : (a) 90 (b) 50 (c) 40 (d) If APB and CQD are two parae ines, then the bisectors of the anges APQ, BPQ, CQP and PQD form : (a) a square (b) a rhombus (c) a rectange (d) any other paraeogram 16. The figure obtained by joining the mid-points of the sides of a rhombus, taken in order, is : (a) a rhombus (b) a rectange (c) a square (d) any paraeogram (a) 50 (b) 55 (c) 80 (d) In a paraeogram ABCD, bisectors of two adjacent anges A and B meet at O. The measure of the ange AOB is equa to : (a) 90 (b) 180 (c) 60 (d) Lengths of two adjacent sides of a paraeogram are in the ratio 2 : 7. If its perimeter is 180 cm, then the adjacent sides of the paraeogram are : (a) 10 cm, 20 cm (b) 20 cm, 70 cm (c) 41 cm, 140 cm (d) none of these 22. If a, b, c and d are four anges of a quadriatera such that a = 2b, b = 2c and c = 2d, then the vaue of d is : (a) 36 (b) 24 (c) 30 (d) none of these 23. The triange formed by joining the mid points of the sides of a right anged triange is : (a) an acute anged triange (b) an obtuse anged triange 2

3 (c) a right anged triange (d) none of these 24. The diagonas of a rectange PQRS intersect at O. If ROQ = 60, then OSP is equa to : (a) 90 (b) 120 (c) 60 (d) none of these 25. In the ABC, B is a right ange, D and E are the mid-points of the sides AB and AC respectivey. If AB = 6 cm and AC = 10 cm, then the ength of DE is : 29. If one ange of a paraeogram is 56 more than three times of its adjacent ange, then measures of a the anges are : (a) 31, 149, 31, 149 (b) 59, 121, 59, 121 (c) 37, 143, 37, 143 (d) none of these 30. In a trapezium ABCD, AB CD, A = (2x 35 ), B = y, C = 85 and D = (3x + 65 ). The vaues of x and y are respectivey : (a) 30, 60 (b) 45, 75 (c) 75, 115 (d) 30, In the ABC, E and F are the mid-points of AB and AC respectivey. The atitude AP intersects EF at Q. The correct reation between AQ and QP is : SHAN (a) 3 cm (b) 5 cm (c) 4 cm (d) 6 cm 26. In a ABC, D, E and F are respectivey the midpoints of BC, CA and AB as shown in the figure. The perimeter of DEF is : 1 (a) (AB + BC + CA) (b) AB + BC + CA 2 (c) 2 (AB + BC + CA) (d) none of these 27. In a paraeogram ABCD, A = (3x + 15 ) and B = (5x 35 ).The measure of D is : (a) 125 (b) 90 (c) 180 (d) cannot be determined 28. In the given figure, if ABCD is a paraeogram, then the vaue of 2 ABC ADC is : (a) 40 (b) 220 (c) 70 (d) 75 3 (a) AQ > QP (c) AQ < QP (b) AQ = QP (d) none of these 32. The quadriatera formed by joining the mid-points of the sides of a quadriatera PQRS, taken in order, is a rectange, if : (a) PQRS is a rectange (b) PQRS is a paraeogram (c) diagonas of PQRS are perpendicuar (d) diagonas of PQRS are equa 33. The quadriatera formed by joining the mid-points of the sides of a quadriatera PQRS, taken in order, is a rhombus, if : (a) PQRS is a rhombus (b) PQRS is a paraeogram (c) diagonas of PQRS are perpendicuar (d) diagonas of PQRS are equa 34. D and E are the mid-points of the sides AB and AC respectivey of ABC. DE is produced to F. To prove that CF is equa and parae to DA, we need an additiona information which is : (a) DAE = EFC (b) AE = EF (c) DE = EF (d) ADE = ECF 35. D and E are the mid-points of the sides AB and AC of ABC and O is any point on side BC. O is joined to A. If P and Q are the mid-points of OB and OC respectivey, then DEQP is : (a) a square (b) a rectange (c) a rhombus (d) a paraeogram

4 36. The figure formed by joining mid-points of the sides of a quadriatera ABCD, taken in order, is a square ony if : (a) ABCD is a rhombus (b) diagonas of ABCD are equa (c) diagonas of ABCD are equa and perpendicuar (d) diagonas of ABCD are perpendicuar. B. Questions From CBSE Examination Papers 1. ABCD is a rhombus such that ACB = 40, then ADC is : (a) 40 (b) 45 (c) 100 (d) If the anges of a quadriatera ABCD, taken in order, are in the ratio 3 : 7 : 6 : 4, then ABCD is a : (a) rhombus (c) paraeogram (b) kite (d) trapezium 3. Two adjacent anges of a rhombus are 3x 40 and 2x The measurement of the greater ange is : (a) 160 (b) 100 (c) 80 (d) In a paraeogram ABCD, A = 60, then D is equa to : (a) 110 (b) 140 (c) 120 (d) In the figure, ABCD is a paraeogram. If DAB = 60 and DBC = 80, then CDB is : (a) 40 (b) 80 (c) 60 (d) In the figure, ABCD is a paraeogram. If B = 100, then ( A + C) is equa to : ABCD is a paraeogram in which DAC = 40 ; BAC = 30 ; DOC = 105 then CDO equas : (a) 75 (b) 70 (c) 45 (d) Anges of a quadriatera are in the ratio 3 : 6 : 8 : 13. The argest ange is : (a) 178 (b) 90 (c) 156 (d) ABCD is a rhombus such that one of its diagonas is equa to its side. Then the anges of rhombus ABCD are : (a) 45, 135, 45, 135 (b) 100, 80, 100, 80 (c) 120, 60, 120, 60 (d) 60, 60, 60, D, E, F are midpoints of sides BC, CA and AB of ABC. If perimeter of ABC is 12.8 cm, then perimeter of DEF is : (a) 17 cm (b) 38.4 cm (c) 25.6 cm (d) 6.4 cm 8. In a quadriatera three anges are in the ratio 3 : 3 : 1 and one of the anges is 80, then other anges are : (a) 120, 120, 40 (b) 100, 100, 80 (c) 110, 110, 60 (d) 90, 90, Two adjacent anges of a paraeogram are (2x + 30) and (3x + 30). The vaue of x is : (a) 30 (b) 60 (c) 24 (d) In a quadriatera ABCD, AB = BC and CD = DA, then the quadriatera is a : (a) paraeogram (b) rhombus (c) kite (d) trapezium 4 (a) 360 (b) 200 (c) 180 (d) A the anges of a convex quadriatera are congruent. However, not a its sides are congruent. What type of quadriatera is it? (a) paraeogram (b) square (c) rectange (d) trapezium 15. ABCD is a quadriatera and AP and DP are bisectors of A and D. The vaue of x is : (a) 60 (b) 85 (c) 95 (d) In a PQR, right anged at Q, PQ = 24 cm and QR = 7 cm. S is the mid point of PR. Then RS is : (a) 3.5 cm (b) 12 cm (c) 25 cm (d) 12.5 cm

5 17. If C = D = 50, then four points A, B, C, D : (a) are concycic (b) do not ie on same circe (c) are coinear (d) A, B, D and A, B, C ie on different circes 18. In paraeogram ABCD, if A = 2x + 15, B = 3x 25, then vaue of x is : (a) 91 (b) 89 (c) 34 (d) Three anges of a quadriatera are 70, 120 and 65. The fourth ange of the quadriatera is : (a) 95 (b) 75 (c) 105 (d) If PQRS is a paraeogram, then Q S is equa to : (a) 90 (b) 120 (c) 180 (d) Which of the foowing is not true for a paraeogram? (a) opposite sides are equa (b) opposite anges are equa (c) opposite anges are bisected by diagonas (d) diagonas bisect each other 22. If APB and CQD are parae ines and a transversa PQ cut them at P and Q, then the bisectors of anges APQ, BPQ, CQP and PQD form a (a) rectange (b) rhombus (c) square (d) any other paraeogram 23. ABCD is a rhombus such that ACB = 40, then ADB is : (a) 40 (b) 45 (c) 50 (d) The figure obtained by joining mid-points of adjacent sides of a rectange of sides 8 cm and 6 cm is : (a) a rectange of area 24 cm 2 (b) a square of area 25 cm 2 (c) a trapezium of area 24 cm 2 (d) a rhombus of area 24 cm If the diagonas AC and BD of a quadriatera ABCD bisect each other, then ABCD is a : (a) paraeogram (b) rectange (c) rhombus (d) trapezium 26. The diagonas of a paraeogram PQRS intersect at O. If QOR = 90 and QSR = 50, then ORS is : (a) 90 (b) 40 (c) 70 (d) If in a quadriatera ABCD, A = 90 and AB = BC = CD = DA, then ABCD is a (a) a paraeogram (b) rectange (c) square (d) rhombus 28. In qudriatera PQRS, if P = 60 and Q : R : S = 2 : 3 : 7, then S is : (a) 175 (b) 135 (c) 150 (d) A quadriatera whose diagonas are equa and bisect each other at right anges is a : (a) rhombus (b) square (c) trapezium (d) rectange 30. In a paraeogram ABCD, if A = 75, then B is : (a) 75 (b) 105 (c) 15 (d) In the given figure, ABCD is a rhombus in which diagonas AC and BD intersect at O. Then AOB is : AS (a) 60 (b) 80 (c) 90 (d) The diagonas AC and BD of a paraeogram ABCD inntersect each other at the point O. If DAC = 32, AOB = 70, then DBC is equa to : (a) 24 (b) 88 (c) 38 (d) Which of the foowing is not a paraeogram? (a) rhombus (b) rectange (c) trapezium (d) square 34. Two anges of a quadriatera are 50 and 80 and other two anges are in the ratio 8 : 15, then the remaining two anges are : (a) 140, 90 (b) 100, 130 (c) 80, 150 (d) 70, In a quadriatera ABCD, if A = 80, B = 70, C = 130, then D is : (a) 80 (b) 70 (c) 130 (d) In an equiatera triange ABC, D and E are the mid points of sides AB and AC respectivey. Then ength of DE is : (a) not possibe to find (b) 3 cm (c) 1 2 BC (d) 3 2 BC 5

6 37. In a quadriatera ABCD, diagonas bisect each other at right anges. Aso, AB = BC = AD = 6 cm, then ength of CD is : (a) 3 cm (b) 6 cm (c) 6 2 cm (d) 12 cm 38. In the figure, PQRS is a paraeogram in which PSR = 125, RQT is equa to : the ratio 1 : 3, then the smaer ange is : (a) 50 (b) 90 (c) 60 (d) In the figure, PQRS is a rectange. If RPQ = 30, then the vaue of (x + y) is : (a) 75 (b) 65 (c) 55 (d) Two consecutive anges of a paraeogram are in (a) 90 (b) 120 (c) 150 (d) The diagonas of a rhombus are 12 cm and 16 cm. The ength of the side of the rhombus is : (a) 10 cm (b) 12 cm (c) 16 cm (d) 8 cm SHORT ANSWER TYPE QUESTIONS 1. Can 95, 70, 110 and 80 be the anges of a quadriatera? Why or why not? 2. Three anges of a quadriatera are equa. Is it a paraeogram? 3. Diagonas of a paraeogram are perpendicuar to each other. Is this statement true? Give reason for your answer. 4. Diagonas of a quadriatera PQRS bisect each other. If P = 35, find Q. 5. The anges of a quadriatera are in the ratio 2 : 4 : 5 : 7. Find the anges. 6. The engths of the diagonas of a rhombus are 24 cm and 18 cm. Find the ength of each side of YAL B A. Important Questions [2 Marks] the rhombus. 7. Diagonas AC and BD of a paraeogram ABCD intersect each other at O. If OA = 3 cm and OD = 2 cm, find the engths of AC and BD. 8. Can a the anges of a quadriatera be acute? Give reason for your answer. 9. In ABC, P, Q and R are mid-points of sides BC, CA and AB respectivey. If AC = 21 cm, BC = 29 cm and AB = 30 cm, find the perimeter of the quadriatera ARPQ. 10. Diagonas of a quadriatera ABCD bisect each other. If A = 35, then B = 145. Is it true? Justify your answer. B. Questions From CBSE Examination Papers 1. The sides BA and DC of quadriatera ABCD are produced as shown in the figure. Prove that x + y = a + b. 2. In the figure, ABCD is a square. A ine segment DX cuts the side BC at X and the diagona AC at O such that COD = 105. Find the vaue of x. 3. In ABC, D and E are mid points of AB and AC. If AD = 3.5 cm; AE = 4 cm ; DE = 2.5 cm, find the perimeter of ABC. 4. ABCD is a rhombus in which AC = 16 cm ; BC = 10 cm. Find the ength of the diagona BD. 5. In ABC, AB = 12 cm, BC = 15 cm and AC = 7 cm. Find the perimeter of the triange formed by joining the mid points of the sides of the triange. 6. ABCD is a rhombus. AO = 5 cm. Area of the rhombus is 25 sq cm. Find the ength of BD. 6

7 7. ABCD is a paraeogram. L and M are points on AB and DC respectivey such that AL = MC. Prove that LM and BD bisect each other. 8., m and n are three parae ines intersected by transversa p and q such that, m and n cut equa intercepts AB and BC on p. Show that, m, n cut off equa intercepts DE and EF on q aso. 13. In ABC, B = 90, D and E are the mid-points of th e sides AB an d AC respectivey. If AB = 6 cm and AC = 10 cm, then find the ength of DE. 14. Anges of a quadriatera are in the ratio 3 : 4 : 5 : 6. Find the anges of the quadriatera. 15. In ABC, AD is the median. A ine through D and parae to AB, meets AC at E. Prove that BE is the median of triange ABC. 16. ABCD is paraeogram. The ange bisectors of A and D intersect at O. Find the measures of AOD. 17. In the given figure, PQRS is a paraeogram and ine segments PA and RB bisect the anges P and R respectivey. Show that PA RB. 9. In the figure, ABCD is a paraeogram. Find the measure of the anges x, y. 10. In the figure, ABCD is a paraeogram. Compute DCA, ACB and ADC, given DAC = 60 and ABC = 75. SHAN 18. In ABC, D, E and F are mid points of sides AB, BC and CA. Show that ABC is divided into four congruent trianges by joining D, E and F. 19. ABCD is a paraeogram and AP and CQ are perpendicuars from vertices A and C on diagona BD. Show that : (i) APB CQD (ii) AP = CQ 11. Prove that if the diagonas of a quadriatera bisect each other, then it is a paraeogram. 12. In the figure, ABCD is a paraeogram. BA is produced to E such that AE = AD. ED is produced to meet BC produced at F. Show that CD = CF. 20. If anges of a quadriatera are in the ratio 1 : 2 : 3 : 4, find measures of a the anges of the quadriatera. 21. Two opposite anges of a paraeogram are (3x 2) and (63 2x). Find a the anges of the paraeogram. 22. Prove that diagona of a paraeogram divides it into two congruent trianges. 23. ABCD is a quadriatera in which P, Q, R and S are mid points of AB, BC, CD and DA respectivey. Show that PQRS is a paraeogram. 7

8 24. If PQRS is a rhombus with PQR = 55, find PRS. 25. D and E are the mid-points of sides AB and AC respectivey of triange ABC. If the perimeter of ABC = 35 cm, find the perimeter of ADE. 26. The vertices of a paraeogram ie on a circe. Prove that its diagonas are equa. 27. The anges of a quadriatera are in the ratio 3 : 5 : 7 : 9. Find the anges of the quadriatera. 28. In the figure, it is given that BDEF and FDCE are paraeograms. Show that BD = CD. 29. In a cycic quadriatera PQRS, if P R = 50, then find the measure of P and R. 30. In ABC, AD is the median and DE AB. Prove that BE is another median. 31. Show that diagonas of a square are equa and bisect each other at right anges. 32. ABC is a triange right anged at C. A ine through the mid point M of hypotenuse AB and parae to BC intersects AC at D. Show that (i) D is the mid point of AC (ii) MD AC 33. The two opposite anges of a paraeogram are (3x 10) and (2x + 35). Find the measure of a the four anges of the paraeogram. L B AS 34. In ABC, AB = 5 cm, BC = 8 cm and AC = 7 cm. If D and E are respectivey mid-points of AB and BC, determine the ength of DE. Give reasons. 35. In the figure, it is given that BDEF and FDCE are paraeograms. If BD = 4 cm, determine CD. 36. In a paraeogram PQRS, if P = (3x 5), Q = (2x + 15), find the vaue of x. 37. In the figure, PQRS is a paraeogram. Find the vaues of x and y. 38. In th e figur e, AOB = 90, AC = BC, OA = 12 cm and OC = 6.5 cm. Find the measure of OB. 39. The anges of a quadriatera are in the ratio 3 : 5 : 9 : 13. Find a the anges of the quadriatera. 40. The ange between the two atitudes of a paraeogram through the vertex of an obtuse ange is 50. Find the anges of the paraeogram. 41. In a paraeogram, show that the ange bisectors of two adjacent anges intersect at right ange. 42. Find the measure of each ange of a paraeogram, if one of its anges is 30 ess than twice the smaer ange. 43. ABCD is a rhombus with ABC = 58. Find ACD. 8

9 SHORT ANSWER TYPE QUESTIONS [3 Marks] A. Important Questions 1. ABCD is a rhombus in which atitude from D on side AB bisects AB. Find the anges of the rhombus. 2. In a paraeogram sh ow th at th e ange bisectors of two adjacent anges intersect at right anges. 3. One ange of a quadriatera is 108 and the remaining three anges are equa. Find each of the three equa anges. 4. E and F are points on diagona AC of a paraeogram ABCD such that AE = CF. Show that BFDE is a paraeogram. 5. D, E and F are the mid-points of the sides BC, CA and AB respectivey of an equiatera ABC. Show that DEF is aso an equiatera triange. 6. In a triange ABC, median AD is produced to X such that AD = DX. Prove that ABXC is a paraeogram. 7. In the figure, through A, B and C ines RQ, PQ and PR have been drawn respectivey parae to sides BC, CA and AB of a ABC. Show that 8. Show that the quadriatera formed by joining the mid-points of sides of a square is aso a square. 9. E is the mid-point of the side AD of trapezium ABCD with AB DC. A ine through E drawn parae to AB intersects BC at F. Show that F is the mid-point of BC. 10. If one ange of a paraeogram is a right ange, it is a rectange. Prove. 11. In the figure, P is the mid-point of side BC of a paraeogram ABCD such that BAP = DAP. Prove that AD = 2CD. BC = 1 2 QR. B. Questions From CBSE Examination Papers 1. Show that bisectors of the anges of a paraeogram form a rectange. 2. Prove that a paraeogram is a rhombus if its diagonas bisect at right anges. 3. Two parae ines and m are intersected by a transversa p. Show that the quadriatera formed by the bisectors of interior anges is a rectange. GOY 5. AD is the median of ABC. E is the midpoint of AD. BE produced meets AC at F. Show that AF = 1 AC In a paraeogram ABCD, bisector of A, aso bisects BC at X. Prove that AD = 2 AB. 9

10 6. Prove that ine segments joining the mid points of opposite sides of any quadriatera bisect each other. 7. Prove that the diagonas of a rectange are equa. 8. Show that if the diagonas of a quadriatera bisect each other at right anges, then it is a rhombus. 9. In a quadriatera ABCD, AO and BO are the bisectors of A and B respectivey. Prove that AOB = 1 ( C + D) In the figure, ABCD is a square, if PQR = 90 and PB = QC = DR, prove that QB = RC, PQ = QR, QPR = ABCD is paraeogram. On diagona BD are points P and Q such that DP = BQ. Show that APCQ is a paraeogram. 18. In the figure, diagona BD of paraeogram ABCD bisects B. Show that it bisects D aso. HAN 11. In the figure, points A and B are on the same side of a ine m, AD m and BE m and meet m at D and E respectivey. If C is the mid point of AB, prove that CD = CE. 19. PQRS is a paraeogram and SPQ = 60. If the bisectors of P and Q meet at point A on RS, prove that A is mid-point of RS. 20. Prove that quadriatera formed by bisectors of the anges of a paraeogram is a rectange. 12. ABCD is a paraeogram in which X and Y are the mid-points of AB and CD. AY and DX are joined which intersect each other at P. BY and CX are aso joined which intersect each other at Q. Show that PXQY is a paraeogram. 13. ABCD is a rectange in which diagona AC bisects A as we as C. Show that (i) ABCD is a square (ii) Diagona BD bisects B as we as D 14. Diagona AC of a paraeogram bisects A. Show that (i) it bisects C aso (ii) ABCD is a rhombus 15. Show that the diagonas of a rhombus are perpendicuar to each other. 16. Show that a diagona of a paraeogram divides it into two congruent trianges and hence prove that the opposite sides of a paraeogram are equa. 21. Show that the ine segments joining the mid-points of opposite sides of quadriatera bisect each other. 22. In the figure, PS and RT are medians of PQR and SM RT. Prove that QM = 1 4 PQ. 10

11 23. In a paraeogram PQRS, the bisectors of adjacent anges R and S intersect each other at the point O. Prove that ROS = In the figure, PQRS is a square. M is the midpoint of PQ and AB RM. Prove that RA = RB. 28. In the figure, DE and BF are perpendicuars to the diagona AC of a paraeogram ABCD. Prove that DE = BF. 25. ABCD is a rhombus. Show that diagona AC bisects A as we as C and diagona BD bisects B as we as D. 26. Prove that the bisectors of any two consecutive anges of a paraeogram intersect at right anges. 27. In the figure, ABCD is a trapezium in which AB DC. E is the mid point of AD and F is a point of BC such that EF DC. Prove that F is the mid point of BC. LONG ANSWER TYPE QUESTIONS 1. In the figure, ABC is an isoscees triange in which AB = AC, CD AB and AD is bisectors of exterior CAE of ABC. Prove that CAD = BCA and ABCD is a paraeogram. 2. PQ and RS are two equa and parae ine segments. Any point M not ying on PQ or RS is joined to Q and S and ines through P parae to QM and through R parae to SM meet at N. A. Important Questions 29. In the figure, PQRS is a paraeogram in which PQ is produced to T such that QT = PQ. Prove that ST bisects RQ. BR ASH [4 Marks] Prove that ine segments MN and PQ are equa and parae to each other. 3. A diagona of a paraeogram bisects one of its anges. Prove that it wi bisect its opposite ange aso. 4. Prove that the ine joining the mid-points of the diagonas of a trapezium is parae to the parae sides of the trapezium. 5. ABCD is a rectange in which diagona BD bisects B. Show that ABCD is a square. 6. If ABCD is a trapezium in which AB CD and AD = BC, prove that A = B. 7. P, Q, R, and S are respectivey the mid-points of the sides AB, BC, CD and DA of a quadriatera ABCD in which AD = BC. Prove that PQRS is a rhombus. B. Questions From CBSE Examination Papers 1. In a paraeogram ABCD, E and F are the midpoints of sides AB and CD respectivey. Show that the ine segment AF and EC trisects the diagona BD. 11

12 2. ABC is a triange right anged at C. A ine through the mid-point M of hypotenuse AB and parae to BC intersects AC at D. Show that (i) MD AC (ii) D is mid-point of AC bisectors of A and B meet at P, prove that AD = DP, PC = BC, DC = 2AD. (iii) MC = MA = 1 2 AB 3. Prove that a ine segment joining the mid-points of any two sides of a triange is parae and haf of its third side. 4. Prove that a ine passing through mid-point of one non parae side of a trapezium parae to parae sides bisect the other non parae side. 5. In the figure, ABCD is a paraeogram. E is the mid-point of BC. DE and AB when produced meet at F. Prove that AF = 2AB. 6. If X, Y and Z are the mid-points of sides BC, CA and AB of ABC respectivey, prove that AZXY is a paraeogram. 7. ABCD is a trapezium in which AB CD and AD = BC. Show that (i) A = B (ii) C = D (iii) ABC BAD (iv) Diagona AC = diagona BD 9. Show that the quadriatera formed by joining the mid-points of the sides of a rectange is a rhombus. 10. Prove that in a triange, the ine segment joining the mid-points of any two sides is parae to third side and is haf of it. Using the above, if P, Q, R are the mid-points of sides BC, AC and AB of ABC respectivey and if PQ = 2.5 cm, QR = 3 cm, RP = 3.5 cm, find the engths of AB, BC and CA. 11. If PQR and LMN be two trianges given in such a way that PQ LM, PQ = LM, QR = MN and QR MN, then show that PR LN and PR = LN. AKA 12. ABCD is a square and on the side DC, an equiatera triange is constructed. Prove that (i) AE = BE and (ii) DAE = ABCD is a paraeogram in which A = 60. If 13. The diagonas of a quadriatera ABCD are perpendicuar, show that quadriatera formed by joining the mid-points of its sides, is rectange. FORMATIVE ASSESSMENT Activity-1 Objective : To verify the mid-point theorem for a triange using paper cutting and pasting. Materias Required : White sheets of paper, a pair of scissors, coour pencis, guestick, geomety box, etc. 12

13 Procedure : 1. On a white sheet of paper, draw a ABC and cut it out. Figure-1 2. Using paper foding method, find the mid-points of AB, AC and BC and mark them as X, Y and Z respectivey. Join X to Y. Figure-2 3. Cut out the trianguar piece AXY and superimpose AY over YC such that YX fas aong CB as shown beow. Figure-3 RS PRAKA Observations : 1. In figure 3(b), we see that AYX exacty covers YCB or AYX = YCB But, AYX and YCB are corresponding anges made on XY and BC by the transversa AC. Therefore, XY BC. But, X and Y are the mid-points of AB and AC respectivey. 2. In figure 3(b), we aso observe that X and Z coincide. It impies XY = CZ. But, CZ is haf of BC. Thus, we can say that the ine segment joining the mid-points of two sides of a triange is parae to the third side and is equa to haf of it. Concusion : From the above activity, the mid-point theorem is verified. Do Yoursef : Draw a right triange, an acute anged triange and an obtuse anged triange. Verify the mid-point theorem for each case. Activity-2 Objective : To verify that a diagona of a paraeogram divides it into two congruent trianges. Materias Required : White sheets of paper, coour pencis, a pair of scissors, guestick, geometry box, etc. Procedure : 1. On a white sheet of paper, draw a paraeogram ABCD and cut it out. Draw the diagona AC of the paraeogram and cut it aong AC to get two trianguar cut outs. 13

14 Figure-1 2. Now, superimpose one triange over the other as shown beow. Figure-2 Observations : Concusion : Do Yoursef : In figure 2, we see that the two trianges exacty cover each other. Hence, the trianges are congruent. From the above activity, it is verified that a diagona of a paraeogram divides it into two congruent trianges. Draw three different paraeograms and verify the above property by paper cutting and pasting. Activity-3 Objective : To verify that the diagonas of a paraeogram bisect each other. Materias Required : White sheets of paper, coour pencis, guestick, a pair of scissors, geometry box, etc. Procedure : 1. On a white sheet of paper, draw a paraeogram ABCD and both its diagonas AC and BD intersecting at O. Cut out the four trianges so formed. GOYAL Figure-1 14

15 . 2. Superimpose OAB over OCD and OBC over ODA as shown. Figure-2 Observations : In figure 2, we see that OAB exacty covers OCD and OBC exacty covers ODA. Or OAB OCD and OBC ODA. So, OA = OC and OB = OD. Concusion : From the above activity, it is verified that the diagonas of a paraeogram bisect each other. Activity-4 Objective : To show that the figure obtained by joining the mid points of consecutive sides of a quadriatera is a paraeogram. Materias Required : White sheets of paper, a pair of scissors, coour pencis, guestick, geometry box, etc. Procedure : 1. On a white sheet of paper, draw a quadriatera ABCD and cut it out. Figure-1 2. By paper foding, find the mid points of AB, BC, CD and DA and mark them as P, Q, R and S respectivey. Join P to Q, Q to R, R to S and S to P. Figure-2 3. Cut out the quadriatera PQRS. Join PR. Figure-3 15

16 4. Cut the quadriatera PQRS aong PR to get two trianges. Superimpose the trianges PQR and PSR such that PQ fas aong RS as shown. Observations : In figure 4(b), we see that PQR exacty covers RSP. PQ = RS and QR = SP PQRS is a paraeogram. [ Each pair of opposite sides of the quadriatera are equa.] Concusion : From the above activity, we can say that the figure obtained by join ing the mid-poin ts of consecutive sides of a quadriatera is a paraeogram. Figure-4 Do Yoursef : Verify the above property by drawing three different quadriateras. 16