**Exercise 13.1**

**1. State which pairs of triangles in the figure given below are similar. Write the similarity rule used and also write the pairs of similar triangles in symbolic form (all lengths of sides are in cm):**

**Solution:-**

**Solution:-**

**4. It is given that ∆ABC ~ ∆EDF such that AB = 5 cm,AC = 7 cm,DF = 15 cm and DE = 12 cm.**

**Find the lengths of the remaining sides of the triangles.**

**Solution:-**

**5. (a) If ∆ABC ~ ∆DEF,AB = 4 cm,DE = 6 cm,EF = 9 cm and FD = 12 cm, then find the perimeter of ∆ABC.**

**Solution:-**

**(b) If ∆ABC ~ ∆PQR, Perimeter of ∆ABC = 32 cm, perimeter of ∆PQR = 48 cm and PR = 6 cm, then find the length of AC.**

**Solution:-**

**6. Calculate the other sides of a triangle whose shortest side is 6 cm and which is similar to a triangle whose sides are 4 cm,7 cm and 8 cm.**

**Solution:-**

**7.**

**(a) In the figure given below, AB || DE,AC = 3 cm,CE = 7.5 cm and BD = 14 cm. Calculate CB and DC.**

**Solution:-**

**(b) In the figure (2) given below, CA || BD, the lines AB and CD meet at G.**

**(i) Prove that ∆ACO ~ ∆BDO.**

**(ii) If BD = 2.4 cm,OD = 4 cm,OB = 3.2 cm and AC = 3.6 cm, calculate OA and OC.**

**Solution:-**

**8. (a) In the figure**

**(i) given below, ∠P = ∠RTS.**

**Prove that ∆RPQ ~ ∆RTS.**

**Solution:-**

**(b) In the figure(2) given below,**

**<ADE=<ACB.**

**(i) Prove that ∆s ABC and AED are similar.**

**(ii) If AE=3cm,BD=1cm and AB=6cm, calculate AC.**

**Solution:-**

**(c) In the figure (3) given below, ∠PQR = ∠PRS. Prove that triangles PQR and PRS are similar. If PR = 8 cm,PS = 4 cm, calculate PQ.**

**Solution:-**

**10. In the given figure, ABC is a triangle in which AB = AC.P is a point on the side BC such that PM ⊥ AB and PN ⊥ AC. Prove that BM x NP = CN× MP.**

**Solution:-**

**11. Prove that the ratio of the perimeters of two similar triangles is the same as the ratio of their corresponding sides.**

**Solution:-**

**12. In the adjoining figure, ABCD is a trapezium in which AB || DC. The diagonals AC and BD intersect at O. Prove that AO/OC = BO/OD**

**Using the above result, find the values of x if OA = 3x – 19,OB = x – 4,OC = x – 3 and OD = 4.**

**Solution:-**

^{2 }– 4x – 3x + 12 = 12x – 76

^{2}– 7x + 12 – 12x + 76 = 0

^{2}– 19x + 88 = 0

^{2}– 8x – 11x + 88 = 0

**15. (a) In the figure (1) given below, E is a point on the side AD produced of a parallelogram ABCD and BE intersects CD at F. show that ∆ABE ~ ∆CFB.**

**Solution:-**

**(b) In the figure (2) given below, PQRS is a parallelogram; PQ = 16 cm,QR = 10 cm.L is a point on PR such that RL∶ LP = 2∶ 3.QL produced meets RS at M and PS produced at N.**

**(i) Prove that triangle RLQ is similar to triangle PLN. Hence, find PN.**

**Solution:-**

**(ii) Name a triangle similar to triangle RLM. Evaluate RM.**

**16. The altitude BN and CM of ∆ABC meet at H. Prove that**

**(i) CN × HM = BM × HN**

**17. In the given figure, CM and RN are respectively the medians of ∆ABC and ∆PQR. If ∆ABC ~ ∆PQR, prove that:**

**(i) ∆AMC ~ ∆PQR**

**(ii) CM/RN = AB/PQ**

**(iii) ∆CMB ~ ∆RNQ**

**Solution:-**

**18. In the adjoining figure, medians AD and BE of ∆ABC meet at the point G, and DF is drawn parallel to BE. Prove that**

**(i) EF = FC**

**(ii) AG∶ GD = 2∶ 1**

**Solution:-**

**19.(a) In the figure given below, AB,EF and CD are parallel lines. Given that AB =15 cm,EG = 5 cm,GC = 10 cm and DC = 18 cm. Calculate**

**(i) EF**

**(ii) AC.**

**Solution:-**

**(b) In the figure given below, AF,BE and CD are parallel lines. Given that AF = 7.5 cm,CD = 4.5 cm,ED = 3 cm,BE = x and AE = y. Find the values of x and y.**

**Solution:-**

**21. In the given figure, ∠A = 90° and AD ⊥ BC If BD = 2 cm and CD = 8 cm, find AD.**

**Solution:-**

**22. A 15 metres high tower casts a shadow of 24 metres long at a certain time and at the same time, a telephone pole casts a shadow 16 metres long. Find the height of the telephone pole.**

**Solution:-**

**23. A street light bulb is fixed on a pole 6 m above the level of street. If a woman of height casts a shadow of 3 m, find how far she is away from the base of the pole?**

**Solution:-**

**Exercise 13.2**

**1. (a) In the figure (i) given below if DE || BG,AD = 3 cm,BD = 4 cm and BC = 5 cm. Find (i) AE∶ EC (ii) DE.**

**Solution:-**

**(b) In the figure (ii) given below, PQ || AC,AP = 4 cm,PB = 6 cm and BC = 8 cm. Find CQ and BQ.**

**Solution:-**

**(c) In the figure (iii) given below, if XY || QR,PX = 1 cm,QX = 3 cm,YR = 4.5 cm and QR = 9 cm, find PY and XY.**

**Solution:-**

**2. In the given figure, DE || BC.**

**(i) If AD = x,DB = x – 2,AE = x + 2 and EC = x – 1, find the value of x.**

**(ii) If DB = x – 3,AB = 2x,EC = x – 2 and AC = 2x + 3, find the value of x.**

**Solution:-**

^{2}– x = x

^{2}– 4

^{2}– 4x = 2x

^{2}– 6x + 3x – 9

^{2}– 4x – 2x

^{2}+ 6x – 3x = -9

**3. E and F are points on the sides PQ and PR respectively of a ∆PQR. For each of the following cases, state whether EF || QR:**

**(i) PE = 3.9 cm,EQ = 3 cm,PF = 8 cm and RF = 9 cm.**

**Solution:-**

**(ii) PQ = 1.28 cm,PR = 2.56 cm,PE = 0.18 cm and PF = 0.36 cm.**

**Solution:-**

**4. A and B are respectively the points on the sides PQ and PR of a triangle PQR such that PQ = 12.5 cm,PA = 5 cm,BR = 6 cm and PB = 4 cm. Is AB || QR? Give reasons for your answer.**

**Solution:-**

**5.(a) In figure (i) given below, DE || BC and BD = CE. Prove that ABC is an isosceles triangle.**

**Solution:-**

**(b) In figure (ii) given below, AB || DE and BD || EF. Prove that DC² = CF × AC.**

**Solution:-**

^{2}= CF× AC

**6.(a) In the figure (i) given below, CD || LA and DE || AC. Find the length of CL if BE = 4 cm and EC = 2 cm.**

**Solution:-**

**(b) In the give figure, ∠D = ∠E and AD/BD = AE/EC. Prove that BAC is an isosceles triangle.**

**Solution:-**

**7. In the adjoining given below, A,B and C are points on OP,OQ and OR respectively such that AB || PQ and AC || PR. show that BC || QR.**

**Solution:-**

**8. ABCD is a trapezium in which AB || DC and its diagonals intersect each other at O. Using Basic Proportionality theorem, prove that AO/BO = CO/DO**

**Solution:-**

**9.(a) In the figure (1) given below, AB || CR and LM || QR.**

**(i) Prove that BM/MC = AL/LQ**

**(ii) Calculate LM∶ QR, given that BM∶ MC = 1∶ 2.**

**Solution:-**

**(b) In the figure (2) given below AD is bisector of ∠BAC. If AB = 6 cm,AC = 4 cm and BD = 3cm, find BC**

**Solution:-**

**Exercise 13.3**

**1. Given that ∆s ABC and PQR are similar.**

**Find:**

**(i) The ratio of the area of ∆ABC to the area of ∆PQR if their corresponding sides are in the ratio 1∶ 3.**

**(ii) the ratio of their corresponding sides if area of ∆ABC : area of ∆PQR = 25∶ 36.**

**Solution:-**

^{2}/QR

^{2}

^{2}/3

^{2}

^{2}/QR

^{2}

^{2}/QR

^{2}= 25/36

^{2}= (5/6)

^{2}

**2. ∆ABC ~ DEF. If area of ∆ABC = 9 sq.cm., area of ∆DEF =16 sq.cm and BC = 2.1 cm., find the length of EF.**

**Solution:-**

^{2}/EF

^{2}

^{2}/EF

^{2}

^{2}/EF

^{2}

^{2}/x

^{2}

**3. ∆ABC ~ ∆DEF. If BC = 3 cm,EF = 4 cm and area of ∆ABC = 54 sq.cm. Determine the area of ∆DEF.**

**Solution:-**

^{2}/EF

^{2}

^{2}/4

^{2}

**4. The area of two similar triangles are 36 cm² and 25 cm². If an altitude of the first triangle is 2.4 cm, find the corresponding altitude of the other triangle.**

**Solution:-**

^{2}

^{2}

^{2}/XN

^{2}

^{2}/a

^{2}

^{2}= 25 (2.4)

^{2}

**5.(a) In the figure, (i) given below, PB and QA are perpendiculars to the line segment AB. If PO = 6 cm,QO = 9 cm and the area of ∆POB = 120 cm², find the area of ∆QOA.**

**Solution:-**

^{2}/PO

^{2}

^{2}/6

^{2}

^{2}

**b) In the figure (ii) given below, AB || DC.AO = 10 cm,OC = 5cm,AB = 6.5 cm and OD = 2.8 cm.**

**(i) Prove that ∆OAB ~ ∆OCD.**

**(ii) Find CD and OB.**

**(iii) Find the ratio of areas of ∆OAB and ∆OCD.**

**Solution:-**

^{2}/CD

^{2}= (6.5)

^{2}/(3.25)

^{2}

**6.(a) In the figure (i) given below, DE || BC.If DE = 6 cm,BC = 9 cm and area of ∆ADE = 28 sq.cm, find the area of ∆ABC.**

**Solution:-**

^{2}/(BC)

^{2}

^{2}/(9)

^{2}

^{2}

^{2}

**(b) In the figure (ii) given below, DE || BC and AD∶ DB = 1∶ 2, find the ratio of the areas of ∆ADE and trapezium DBCE.**

**Solution:-**

^{2}/AB

^{2}

^{2}

**7.In the given figure, DE || BC.**

**(i) Prove that ∆ADE and ∆ABC are similar.**

**(ii) Given that AD = ½ BD, calculate DE if BC = 4.5 cm.**

**(iii) If area of ∆ABC = 18cm**

^{2}, find the area of trapezium DBCE

**Solution:-**

^{2}/BC

^{2}

^{2}

^{2}

^{2}= 1/9

^{2}

^{2 }

**8. In the given figure, AB and DE are perpendicular to BC.**

**(i) Prove that ∆ABC ~ ∆DEC**

**(ii) If AB = 6 cm: DE = 4 cm and AC = 15 cm, calculate CD.**

**(iii) Find the ratio of the area of ∆ABC : area of ∆DEC.**

**Solution:-**

^{o}]

^{2}/DE

^{2}

^{2}/4

^{2}

**9. In the adjoining figure, ABC is a triangle. DE is parallel to BC and AD/DB = 3/2,**

**(i) Determine the ratios AD/AB,DE/BC**

**(ii) Prove that ∆DEF is similar to ∆CBF. Hence, find EF/FB.**

**(iii) What is the ratio of the areas of ∆DEF and ∆CBF?**

**Solution:-**

^{2}/BC

^{2}

^{2}/BC

^{2}

^{2}

**10. In ∆ABC,AP∶ PB = 2∶ 3.PO is parallel to BC and is extended to Q so that CQ is parallel to BA. Find:**

**(i) area ∆APO : area ∆ABC.**

**(ii) area ∆APO : area ∆CQO.**

**Solution:-**

^{2}/AB

^{2}

^{2}/(AP + PB)

^{2}

^{2}/(2 + 3)

^{2}

^{2}/CQ

^{2}

^{2}/PB

^{2}

^{2}/3

^{2}

**11.(a) In the figure (i) given below, ABCD is a trapezium in which AB || DC and AB = 2 CD. Determine the ratio of the areas of ∆AOB and ∆COD.**

**Solution:-**

^{2}/CD

^{2}

^{2}/CD

^{2}… [because AB = 2 CD]

^{2}/CD

^{2}

**Solution:-**

^{2}/(AD)

^{2}

^{2}/5

^{2}

^{2}/BC

^{2}

^{2}/AD

^{2}= 9/25

**12. In the adjoining figure, ABCD is a parallelogram. P is a point on BC such that BP∶ PC = 1∶ 2 and DP produced meets AB produced at Q. If area of ∆CPQ = 20 cm², find**

**(i) area of ∆BPQ.**

**(ii) area ∆CDP.**

**(iii) area of parallelogram ABCD.**

**Solution:-**

^{2}

^{2}/BP

^{2}

^{2}/1

^{2}

^{2}

^{2}

^{2}

^{2}

^{2}.

**13. (a) In the figure (i) given below, DE || BC and the ratio of the areas of ∆ADE and trapezium DBCE is 4∶ 5. Find the ratio of DE∶ BC.**

**Solution:-**

^{2}/(BC)

^{2}… [equation (i)]

^{2}/(BC)

^{2}

^{2}/(BC)

^{2}= 4

^{2}/9

^{2}

^{2}/(BC)

^{2}= 2/3

**(b) In the figure (ii) given below, AB || DC and AB = 2 DC. If AD = 3 cm,BC = 4 cm and AD,BC produced meet at E, find (i) ED (ii) BE (iii) area of ∆EDC : area of trapezium ABCD.**

**Solution:-**

^{2}/AB

^{2}

^{2}/(2DC)

^{2}

^{2}/4DC

^{2}

**14. (a) In the figure given below, ABCD is a trapezium in which DC is parallel to AB. If AB = 9 cm,DC = 6 cm and BB = 12 cm., find (i) BP (ii) the ratio of areas of ∆APB and ∆DPC.**

**Solution:-**

^{2}/CD

^{2}

^{2}/6

^{2}

**(b) In the figure given below, ∠ABC = ∠DAC and AB = 8 cm,AC = 4 cm,AD = 5 cm. (i) Prove that ∆ACD is similar to ∆BCA (ii) Find BC and CD (iii) Find the area of ∆ACD : area of ∆ABC.**

**Solution:-**

^{2}/AB

^{2}

**15. ABC is a right angled triangle with ∠ABC = 90°. D is any point on AB and DE is perpendicular to AC. Prove that:**

**(i) ∆ADE ~ ∆ACB.**

**(ii) If AC = 13 cm,BC = 5 cm and AE = 4 cm. Find DE and AD.**

**(iii) Find, area of ∆ADE : area of quadrilateral BCED.**

**Solution:-**

^{o}]

^{2}= AB

^{2}+ BC

^{2}

^{2}= AB

^{2}+ 5

^{2}

^{2}+ 25

^{2}= 169 – 25

^{2}= 144

^{2}

^{2}

**16. Two isosceles triangles have equal vertical angles and their areas are in the ratio 7: 16. Find the ratio of their corresponding height.**

**Solution:-**

^{2}/XN

^{2}… [from corollary of theorem]

^{2}= 7/16

**17. On a map drawn to a scale of 1∶ 250000, a triangular plot of land has the following measurements : AB = 3 cm,BC = 4 cm and ∠ABC = 90°. Calculate**

**(i) the actual length of AB in km.**

**(ii) the area of the plot in sq.km:**

**Solution:-**

^{2}× 6 cm

^{2}

^{2}× 6

^{2}

^{2}

**18. On a map drawn to a scale of 1∶ 25000, a rectangular plot of land, ABCD has the following measurements AB = 12 cm and BG = 16 cm. Calculate:**

**(i) the distance of a diagonal of the plot in km.**

**(ii) the area of the plot in sq.km.**

**Solution:-**

^{2}= AB

^{2}+ BC

^{2}

^{2}+ BC

^{2})

^{2}+ (16)

^{2})

^{2}

^{2}× 192 cm

^{2}

^{2}× 192

^{2}

^{2}

**19. The model of a building is constructed with the scale factor 1∶ 30.**

**(i) If the height of the model is 80 cm, find the actual height of the building in metres.**

**(ii) If the actual volume of a tank at the top of the building is 27 m³, find the volume of the tank on the top of the model.**

**Solution:-**

^{3}

^{3}

^{3}

**20. A model of a ship is made to a scale of 1∶ 200.**

**(i) If the length of the model is 4 m, find the length of the ship.**

**(ii) If the area of the deck of the ship is 160000 m², find the area of the deck of the model.**

**(iii) If the volume of the model is 200 liters, find the volume of the ship in m³. (100 liters = 1 m³)**

**Solution:-**

^{2}

^{3}

^{3}

**Chapter Test**

**1. In the adjoining figure, ∠1 = ∠2 and ∠3 = ∠4. Show that PT x QR = PR × ST.**

**Solution:-**

**Solution:-**

**3.(a) In the figure given below. ∠AED = ∠ABC. Find the values of x and y.**

**Solution:-**

**(b) In the figure given below, CD = ½ AC,B is mid-point of AC and E is mid-point of DF. If BF || AG, prove that :**

**(i) CE || AG**

**(ii) 3 ED = GD**

**Solution:-**

**4. In the adjoining figure, 2 AD = BD, E is mid-point of BD and F is mid-point of AC and EC || BH. Prove that:**

**(i) DF || BH**

**(ii) AH = 3 AF.**

**Solution:-**

**5. In a ∆ABC,D and E are points on the sides AB and AC respectively such that DE || BC. If AD = 2.4 cm, AE = 3.2 cm,DE = 2 cm and BC = 5 cm, find BD and CE.**

**Solution:-**

**6. In a ∆ABC,D and E are points on the sides AB and AC respectively such that AD = 5.7cm,BD = 9.5cm,AE = 3.3cm and AC = 8.8cm. Is DE || BC? Justify your answer.**

**Solution:-**

**7. If the areas of two similar triangles are 360 cm² and 250 cm² and if one side of the first triangle is 8 cm, find the length of the corresponding side of the second triangle.**

**Solution:-**

^{2},QR = 8 cm

^{2}

^{2}/yz

^{2}

^{2}/a

^{2}

^{2}

^{2}= (250 ×64)/360

**8. In the adjoining figure, D is a point on BC such that ∠ABD = ∠CAD. If AB = 5 cm,AC = 3 cm and AD = 4 cm, find**

**(i) BC**

**(ii) DC**

**(iii) area of ∆ACD∶ area of ∆BCA.**

**Solution:-**

^{2}/AB

^{2}

^{2}/5

^{2}

**9. In the adjoining figure, the diagonals of a parallelogram intersect at O.OE is drawn parallel to CB to meet AB at E, find area of ∆AOE : area of parallelogram ABCD.**

**Solution:-**

**10. In the given figure, ABCD is a trapezium in which AB || DC.If 2AB = 3DC, find the ratio of the areas of ∆AOB and ∆COD.**

**Solution:-**

^{2}/DC

^{2}

^{2}/2

^{2}

**11. In the adjoining figure, ABCD is a parallelogram. E is mid-point of BC.DE meets the diagonal AC at O and meet AB (produced) at F. Prove that**

**Solution:-**

^{2}/DO

^{2}

^{2}/2

^{2}

**12. A model of a ship is made to a scale of 1: 250 calculate:**

**(i) The length of the ship, if the length of model is 1.6 m.**

**(ii) The area of the deck of the ship, if the area of the deck of model is 2.4 m**

^{2}.**(iii) The volume of the model, if the volume of the ship is 1 km**

^{3}.**Solution:-**

^{2}

^{2}= 4 m

^{2 }

^{3}

^{3}) × 1 km

^{3}

^{3}× 1000

^{3 }

^{3}

^{3}

^{3}.