Assignments in Mathematics Cass IX (Term ) 9. AREAS OF PARALLELOGRAMS AND TRIANGLES IMPORTANT TERMS, DEFINITIONS AND RESULTS If two figures A and B are congruent, they must have equa areas. Or, if A and B are congruent figures, then ar(a) = ar(b) If a panar region formed by a figure T is made up of two non-overapping panar regions formed by figures P and Q, then ar (T) = ar (P) + ar (Q). Two figures are said to be on the same base and between the same paraes, if they have a common base (side) and the vertices (or the vertex) opposite to the common base of each figure ie on a ine parae to the base. Paraeograms on the same base and between the same paraes are equa in area. SUMMATIVE ASSESSMENT Area of a paraeogram is the product of its any side and the corresponding atitude. Paraeograms on the same base and having equa areas ie between the same paraes. If a paraeogram and a triange are on the same base and between the same paraes, then area of the triange, is haf the area of the paraeogram. Two trianges on the same base and between the same paraes are equa in area. Two trianges having the same base and equa areas ie between the same paraes. Area of a triange is haf the product of its base and the corresponding atitude (or height). A median of a triange divides it into two trianges of equa areas. MULTIPLE CHOICE QUESTIONS [1 Mark] 1. If sum of two parae sides of a trapezium is 15 cm and its area is 30 cm, then the height of the trapezium is : (a) cm (b) cm 6 cm (d) 8 cm. The area of a triange is 36 cm and one of its sides is 9 cm. Then, the ength of the corresponding atitude to the given side is : (a) 8 cm (b) cm 6 cm (d) 9 cm 3. The atitude of a paraeogram is twice the ength of the base and its area is 150 cm. The engths of the base and the atitude respectivey are : (a) 0 cm, 0 cm (b) 35 cm, 70 cm 5 cm, 50 cm (d) 15 cm, 30 cm. In the figure, PQRS is a paraeogram, PM RS and RN PS. If PQ = 1 cm, PM = 6 cm and RN = 8 cm, then the ength of PS is equa to : (a) 18 cm (b) 9 cm cm (d) 1 cm 5. ABCD is a paraeogram one of whose diagonas is AC. Then, which of the foowing is true? A. Important Questions (a) ar (ΔADC) > ar (ΔCBA) (b) ar (ΔADC) = ar (ΔCBA) ar (ΔABC) < ar (ΔADC) (d) none of these 6. The area of a rhombus is 0 cm. If one of its diagonas is 5 cm, then the other diagona is : (a) 8 cm (b) 5 cm cm (d) 10 cm 7. Which of the foowing is true? (a) Area of a triange = Base Atitude (b) Atitude of a triange = Area Base Area Base of triange = Atitude (d) none of these 8. The sum of the engths of bases of a trapezium is 13.5 cm and its area is 5 cm. The atitude of the trapezium is : (a) 9 cm (b) 6 cm 8 cm (d) 1 cm 1
9. Two adjacent sides of a paraeogram are cm and 18 cm. If the distance between the onger sides is 1 cm, then the distance between the shorter sides is : (a) 18 cm (b) 16 cm 9 cm (d) none of these 10. Which of the foowing figures ies on the same base and between the same paraes? (a) ony (i) (b) both (i) and (ii) ony (iii) (d) ony (ii) 11. Area of a rhombus is cm, the product of its diagonas is : (a) 8 cm (b) cm 1 cm (d) none of these 1. Sum of the parae sides of a trapezium is 10 cm and its area is 0 cm. The distance between the parae sides is : (a) 10 cm (b) 8 cm cm (d) cm 13. The area of an isoscees triange, if its base and corresponding atitude are 6 cm and cm respectivey, is : (a) 10 cm (b) cm 1 cm (d) 0 cm 1. The side of an equiatera triange is cm. Its area is : (a) 3 cm (b) 3 cm 16 3 cm (d) 1 3 cm 15. The area of the paraeogram ABCD is : (a) 10 cm (b) 9 cm 1 cm (d) 15 cm (a) ar ( gm ADCF) = ar (rect. EFCD) (b) ar ( gm ABCD) = ar (rect. EFCD) ar ( gm ADCF) = ar (rect. ABCD) (d) none of these 17. If the sum of the parae sides of a trapezium is 7 cm and distance between them is cm, then area of the trapezium is : (a) 8 cm (b) 7 cm 1 cm (d) 1 cm 18. ABCD is a quadriatera whose diagona AC divides it into two parts, equa in area, then ABCD : (a) is a rectange (b) is aways a rhombus is a paraeogram (d) need not be any of (a), (b) or 19. The median of a triange divides it into two : (a) trianges of equa area (b) congruent trianges right trianges (d) isoscees trianges 0. If ABCD is a paraeogram, then which of the foowing is true? (a) (b) (d) ar ( ABD) = ar ( BCD) ar ( ABD) = ar ( ABC) ar ( ABC) = ar ( ACD) a are true 1. In the figure, PQRS and PQLM are paraeogram and X is any point on side QL. The area of PMX is equa to : 16. In the figure, ABCD is a paraeogram and EFCD is a rectange. Now which of the foowing is correct option? (a) area of RQL (b) area of gm PQRS area of SPM (d) 1 area of gm PQLM
. ABC is an isoscees triange with each equa side 5 cm, perimeter 18 cm and height AD = 7 cm. Then, the area of the triange ABC is : (a) ar ( AOB) (b) ar ( COD) ar ( BOC) (d) none of these 8. In the figure, DE BC. Then, which of the foowing reations is true? (a) 30 cm (b) 8 cm 1 cm (d) 36 cm 3. The area of a triange is equa to the area of a rectange whose ength and breadth are 18 cm and 1 cm respectivey. If the base of the triange is cm, then its atitude is : (a) 18 cm (b) cm 36 cm (d) 8 cm. In the given figure, ABC is a triange and AD is one of its medians. The ratio of areas of trianges ABD and ACD respectivey is : (a) : 1 (b) 1 : 1 : 1 (d) 3 : 1 5. If the base of an isoscees triange is 8 cm and one of the equa sides measures 5 cm, then the area of the isoscees triange is : (a) cm (b) 18 cm 1 cm (d) 30 cm 6. In the figure, point D divides the side BC of ABC in the ratio p : q. The ratio between the ar ( ABD) and ar ( ADC) is : p q (a) : p + q p + q BR H (b) p : q q : p (d) none of these 7. In the figure, ABCD is a trapezium in which AB CD and its diagonas AC and BD intersect at O. Now ar ( AOD) is equa to : (a) (b) (d) ar ( ACD) = ar ( BOC) ar ( ACD) = ar ( ABE) ar ( ACD) = ar ( BDE) ar ( ACD) = ar ( CDE) 9. P and Q are any two points ying on the sides CD and AD respectivey of a paraeogram ABCD. Now which of the two trianges have equa area? (a) APD and BPC APB and BQC (b) ABQ and CDQ (d) none of these 30. The figure obtained by joining the mid-points of the adjacent sides of a reactange of sides 8 cm and 6 cm is : (a) a rectange of area cm (b) a square of area 5 cm a trapezium of area cm (d) a rhombus of area cm 31. In the figure, the area of paraeogram ABCD is : (a) AB BM (b) BC BN DC DL (d) AD DL 3. The area of the figure formed by joining the mid-points of the adjacent sides of a rhombus with diagonas 1 cm and 16 cm is : (a) 8 cm (b) 6 cm 96 cm (d) 19 cm 3
B. Questions From CBSE Examination Papers 1. If a triange and a paraeogram are on the same base and between the same paraes, then the ratio of the area of the triange to the area of paraeogram is : (a) 1 : (b) 1 : 3 1 : (d) 1 : 1. The mid point of the sides of a triange ABC aong with any one of the vertices as the fourth point makes a paraeogram whose are is equa to : (a) 17 sq units 3 sq units (b) 176 sq units (d) 86 sq units 8. In the figure, AB DC. Which of the foowing is true about the figure? (a) ar (ABC) 3 (b) 1 ar (ABC) 1 3 ar (ABC) (d) 1 ar (ABC) 3. In ABC, D, E, F are respectivey the mid points of the sides AB, BC and AC. Area of DEF : area of ABC is : (a) : 1 (b) 3 : : 3 (d) 1 :. ABCD is paraeogram and O is mid point of AB. If area of the paraeogram is 7 sq cm, then area of DOC is : (a) ar (AOD) = ar (BOC) (b) ar (AOB) = ar (COD) ar (ADC) = ar (ABC) (d) ar (AOB) = 1 ar (ABCD) 9. A rectange and a rhombus are on the same base and between the same paraes. Then the ratio of their areas is : (a) 1 : 1 (b) 1 : 1 : 3 (d) 1 : 10. ABCD is a tr apezium with parae sides AB = a cm, CD = b cm. E and F are the midpoints of non-parae sides. The ratio of ar (ABFE) and ar (EFCD) is : (a) 158 sq cm (b) 37 sq cm 18.5 sq cm (d) sq cm 5. In the figure, ABCD is a paraeogram. AE DC, CF AD. If AB = 16 cm, AE = 8 cm and CF = 10 cm, then AD equas : (a) 1 cm 1.8 cm L BR (b) 15 cm (d) 15.5 cm 6. AD is the median of a triange ABC. Area of triange ADC = 15 cm, then ar ( ABC) is : (a) 15 cm (b).5 cm 30 cm (d) 37.5 cm 7. ABCD is a paraeogram. O is an interior point. If ar (AOB) + ar (DOC) = 3 sq units, then ar ( gm ABCD) is : (a) a : b (b) (3a + b) : (a + 3b) (a + 3b ) : (3a + b) (d) (a + b : (3a + b) 11. If E, F, G, H are respectivey the mid points of the sides of a paraeogram ABCD, and ar (EFGH) = 0 cm, then ar (paraeogram ABCD) is : (a) 0 cm (b) 0 cm 80 cm (d ) 60 cm 1. AD is th e median of a ABC. Ar ea of ADC = 15 cm, then ar ( ABC) is : (a) 15 cm (b).5 cm 30 cm (d) 37.5 cm 13. In the figure, D is the midpoint of side BC of ABC. and E is the midpoint of AD. Then the area of ABE is :
(a) 1 3 area ( ABC) (b) 1 area ( AEC) 1 area ( BEC) (d) 1 area of ( ABC) 1. Two paraeograms are on the same base and between the same paraes. The ratio of their areas is : (a) 1 : 1 (b) 1 : : 1 (d) 1 : 15. In a paraeogram ABCD, P is a point in its interior. If ar ( gm ABCD) = 18 cm, then [ar ( APD) + ar ( CPB)] is : (a) 9 cm (b) 1 cm 18 cm (d) 15 cm 16. ABCD is a paraeogram. If E and F are mid points of sides AB and CD and diagona AC is joined, then ar (FCBE) : ar (CAB) is : (a) 1 : (b) : 1 1 : 1 (d) 1 : 17. If area of gm ABCD is 80 cm, then ar ( ADP) is : (a) 80 cm (b) 60 cm 50 cm (d) 0 cm 18. In ABC, AD is median of ABC and BE is median of ABD. If ar ( ABE) = 15 cm, then ar ( ABC) is : (a) 60 cm (b) 50 cm 0 cm (d ) 30 cm 19. In ABC, E is the mid-point of median AD. ar ( BED) is : (a) 1 ar ( ABC) (b) B ASH 1 ar ( ABC) 3 1 ar ( ABC) (d ) none of the above 0. If a triange and a square are on the same base and between the same paraes, then the ratio of area of triange to the area of square is : (a) 1 : 3 (b) 1 : 3 : 1 (d ) 1 : 5 1. In the figure, AB DC, then the trianges that have equa area are : (a) ADX, ACX (b) ADX, XCB ACX, XCB (d ) a of the above. D and E are the points on the sides AB and AC respectivey of triange ABC such that DE BC. If area of DBC = 15 cm, then area EBC is : (a) 30 cm (b) 7.5 cm 15 cm (d ) 0 cm 3. In the given figure, PQRS is a paraeogram and PQCD is a rectange, then : (a) ar (PQRS) < ar (PQCD) (b) ar(pqrs) = ar(pqcd) ar(pqrs) > ar(pqcd) (d) none of the above. In the given figure, if ABCD is a paraeogram, then ength of BE is : (a) cm (b) 6 cm 6 cm (d) 8 cm 5. If area of paraeogram ABCD is 5 cm and on the same base CD, a triange BCD is given such that area of BCD = x cm, then the vaue of x is : (a) 5 cm (b) 1.5 cm 15 cm (d) 0 cm 6. A triange and a rhombus are on the same base and between the same paraes. Then the ratio of area of triange to that of rhombus is : (a) 1 : 1 (b) 1 : 1 : 3 (d) 1 : 7. If the base of a paraeogram is 8 cm and its atitude is 5 cm, then its area is equa to : (a) 15 cm (b) 0 cm 0 cm (d ) 10 cm
8. In the figure, if paraeogram ABCD and rectange ABEF are of equa area, then : (d) 30. The diagona of a square is 10 cm. Its area is : (a) 0 cm (b) 5 cm 50 cm (d) 100 cm 31. Two poygons have the same area in figure : (a) perimeter of ABCD = perimeter of ABEF (b) perimeter of ABCD < perimeter of ABEF perimeter of ABCD > perimeter of ABEF (d ) perimeter of ABCD = 1 (perimeter of ABEF) 9. In wh ich of th e foowin g figures, on e quadriatera and one triange, ie on the same base and between the same paraes? (a) (b) (a) (b) (d ) 3. The ength of the diagona of the square is 10 cm. The area of the square is : (a) 0 cm (b) 100 cm 50 cm (d) 70 cm SHORT ANSWERS TYPE QUESTIONS A. Important Questions [ Marks] 1. PQRS is a paraeogram whose area is 180 cm and A is any point on the diagona QS. The area of ASR is 90 cm. Is it true?. P is any point on the median AD of ABC. Show that ar(apb) = ar(acp). 3. In th e figur e, PQRS and EFRS ar e two paraeograms. Is area of MFR equa to 1 area of gm PQRS? O 5. BD is one of the diagonas of a quadriatera ABCD. AM and CN are the perpendicuars from A and C respectivey on BD. Show that ar (ABCD) = 1 BD (AM + CN).. In the figure, ABCD is a quadriatera and BD is one of its diagonas. Show that ABCD is a paraeogram and find its area. 6. Check whether the foowing statement is true. PQRS is a rectange inscribed in a quadrant of a circe of radius 13 cm. A is any point on PQ. If PS = 5 cm, then ar (PAS) = 30 cm. 7. In ABC, O is any point on its median AD. Show that ar( ABO) = ar( ACO). 6
8. ABCD is a paraeogram and X is the mid-point of AB. If ar (AXCD) = cm, then ar(abc) = cm. It is true? 9. In the figure, LM = 3 QR, LM QR and distance between LM and QR is 3 cm. If ength of QR = 6 cm, find the area of LQRM. 1. Show that the segment joining the mid-points of a pair of opposite sides of a paraegoram divides it into two equa paraeograms. 13. In the figure, ABCD and EFGD are two paraegorams and G is the mid-point of CD. Check whether area of ΔPDC is equa to haf of area EFGD. 10. ABC and BDE are two equiatera trianges such that D is mid-point of BC. Show that ar (BDE) = 1 ar (ABC). 11. In paraeogram ABCD, AB = 10 cm. The atitude corresponding to the sides AB and AD are respectivey 7 cm and 8 cm. Find AD. 1. In the figure, PQRS is square and T and U are respectivey the mid-points of PS and QR. Find the area of ΔOTS, if PQ = 8 cm. 15. Each side of a rhombus is 8 cm and its area is 36 cm. Find its atitude. B. Questions From CBSE Examination Papers 1. D, E, F are respectivey the mid point of the sides BC, CA and AB of triange ABC. Show that. ar( DEF) = 1 ar ( ABC).. In the figure, AD is a median of ABC. E is any point on AD. Show that ar ( BED) = ar ( CED). 6. In the given figure, ABCD is a paraeogram and AE DC. If AB is 0 cm and the area of paraeogram ABCD is 80 cm, find AE. 3. PQRS is a trapezium with PQ SR. A ine parae to PR intersects PQ at L and QR at M. Prove that ar ( PSL) = ar ( PRM).. In the figure, E is any point on median AD of a ABC. Show that ar (ABE) = ar (ACE). 7. P and Q are any two points ying on the sides DC and AD respectivey of a paraeogram ABCD. Show that ar ( APB = ar ( BQC) 8. In the figure, ABCD is a trapezium. BC = 17 cm, AB = 16 cm and DC = 8 cm. Find the area of ABCD. 5. Show that the median of a triange divides it into two trianges of equa areas. 7
9. The area of a paraeogram ABCD is 0 sq. cm. If X be the mid point of AD, find area of AXB. 10. Diagonas AC and BD of a trapezium ABCD with AD CD intersect each other at O. Prove that ar ( AOD) = ar ( BOC). SHORT ANSWERS TYPE QUESTIONS 1. If the mid-points of the sides of a quadriatera are joined in order, prove that the area of the paraeogram so formed wi be haf of that of the given quadriatera.. In the figure, ABCD is a paraeogram and BC is produced to a point Q such that AD = CQ. If AQ intersects DC at P, show that ar (BPC) = ar (DPQ). A. Important Questions [3 Marks] 8. In the figure, M, N are points on sides PQ and PR respectivey of ΔPQR, such that ar (ΔQRN) = ar (ΔQRM). Show that MN QR. 9. In the figure, O is any point on the diagona PR of a paraeogram PQRS. Show that ar (PSO) = ar (PQO). 3. O is any point on the diagona BD of a paraeogram ABCD. Show that ar (ΔOAB) = ar (ΔOBC).. D is the mid-point of side BC of a ΔABC and E is the mid point of BD. If O is the mid-point of AE, then show that ar (BOE) = 1 ar (ABC). 8 5. In the figure, PSDA is a paraeogram. Points Q and R are taken on PS such that PQ = QR = RS and PA QB RC. Prove that ar (PQE) = ar (CFD). 10. Show that the area of a rhombus is haf the product of the engths of its diagonas. 11. Trianges ABC and DBC are on the same base BC with vertices A and D on opposite sides of BC such that ar (ABC) = ar (DBC). Show that BC bisects AD. 1. In the figure, ABCD is a paraeogram in which BC is produced to E such that CE = BC. AE intersects CD at F. If ar (ΔDFB) = 3 cm, find the area of the paraeogram ABCD. 6. Show that the diagonas of a paraeogram divide it into four trianges of equa area. 7. In the figure, ABCD is a square. E and F are respectivey the mid-points of BC and CD. If R is the mid point of EF, show that ar (AER) = ar (AFR). 13. ABCD is a trapezium with parae sides AB = a cm and DC = b cm. E and F are the mid-points of non-parae sides. Show that ar (ABFE) : ar (EFCD) = (3a + b) : (a + 3b). 8
1. In the figure, ABCD is a trapezium in which AB DC and L is the mid-point of BC. Through L, a ine PQ AD has been drawn which meets AB in P and DC produced to Q. Show that ar (ABCD) = ar (APQD). B. Questions From CBSE Examination Papers 1. XY is a ine parae to side BC of a ABC. If BE AC and CF AB meet XY at E and F respectivey, show that ar ( ABE) = ar ( ACF).. Prove that a rectange and a paraeogram on the same base and between the same paraes, the perimeter of the paraeogram is greater than the perimeter of the rectange. 3. If medians of a triange ABC intersects at G, show that ar ( AGB) = ar ( AGC) 6. In the figure, AD is median. Prove that ar ( ABD) = ar ( ACD). 7. In the figure, ABCD is a quadriatera. A ine through D parae to AC meets BC produced at E. Prove that ar ( ABE) = ar quad. (ABCD). = ar ( BGC) = 1 3 ar ( ABC).. In the figure, diagonas AC and BD of quadriatera ABCD intersect at O such that OB = CD. If AB = CD, then show that ar ( DOC) = ar ( AOB). 8. ABCD is a paraeogram in which CD = 15 cm, its corresponding atitude AM is 8 cm and CN AD. If CN = 10 cm, then find the ength of AD. 5. In the figure, ar (DRC) = ar (DPC) and ar (BDP) = ar (ARC). Show that both the quadriateras ABCD and DCPR are trapeziums. 9. A point D is taken on the base BC of a ABC and AD is produced to E, such that DE = AD. Show that ar ( BCE) = ar ( ABC). 10. ABCD is a paraeogram whose diagonas AC and BD intersect at O. A ine through O intersect AB at P and DC at Q. Prove that : ar ( POA) = ar ( QOC) LONG ANSWERS TYPE QUESTIONS [ Marks] A. Important Questions 1. The medians BE and CF of a triange ABC intersect at G. Prove that the area of ΔGBC = area of the quadriatera AFGE.. D, E, F are the mid-points of the sides BC, CA and AB respectivey of ΔABC. Prove that BDEF is a paraeogram whose area is haf that of ABC. 9
3. In the figure, ABCD is a paraeogram. Points P and Q on BC trisect BC. Show that ar (APQ) = ar (DPQ) = 1 6 ar (ABCD).. In ABC, if L and M are the points on AB and AC respectivey, such that LM BC, prove that ar (LOB) = ar (MOC). 5. In the figure, ABCD and AEFD are two paraeograms. Prove that ar (PEA) = ar (QFD). B. Questions From CBSE Examination Papers 1. The side AB of a paraeogram ABCD is produced to any point P. A ine through A and parae to CP meets CB produced at Q and then paraeogram PBQR is competed. Show that ar (ABCD) = ar (PBQR). 5. The figure, ABCDE is a pentagon and a ine through B parae to AC meets DC produced at F. Show that (i) ar (ACB) = ar (ACF) (ii) ar (ABCDE) = ar (AEDF).. In the figure, ABC is a triange, D is the mid-point of AB, P is any point on BC. Line CQ is drawn parae to PD to intersect AB at Q. PQ is joined. Show that ar ( BPQ) = 1 ar ( ABC). 6. ABCD is a trapezium with AB DC. A ine parae to AC intersects AB at X and BC at Y. Prove that ar (ADX) = ar (ACY). 7. Diagonas of a paraeogram ABCD intersect at point O. Thourgh O, a ine is drawn to intersect AD at P and BC at Q. Show that PQ divides the paraeogram into two parts of equa area. 8. Diagonas PR and QS of quadriatera PQRS intersect at T such that PT = TR. If PS = QR, show that ar ( PTS) = ar ( RTQ). 3. Prove that paraeograms on the same base and between the same paraes are equa in area.. In the figure, ABCD is a paraeogram in which BC is produced to E such that CE = BC. AE intersects CD at F. Show that ar ( BDF) = 1 ar (ABCD). 9. Diagona AC and BD of a quadriatera ABCD intersect at O in such a way that ar (AOD) = ar (BOC). Prove that ABCD is a trapezium. 10. PQRS and ABRS are paraeograms and X is any point on side BR. Show that : (i) area PQRS = area ABRS (ii) area AXS = 1 area PQRS. 11. In the given figure, AP BQ CR. Prove that ar ( AQC) = ar ( PBR). 10
1. A point E is taken as the midpoint of the side BC of a paraeogram ABCD. AE and DC are produced to meet at F. Prove that ar ( ADF) = ar (ABFC). 13. In the figure, M is a point in the interior of a paraeogram PQRS. Show that (i) ar( PMQ) + ar ( MRS) = 1 ar( gm PQRS) (ii) ar( PMS) + ar ( MQR) = ar( PMQ) + ar ( MRS). 1. In the figure, diagonas AC and BD of quadriatera ABCD intersect at O, such that OB = OD. If AB = CD, show that (i) ar (DOC) = ar (AOB) (ii) ar (DCB) = ar (ACB) (iii) ABCD is a paraeogram. 11