M l aggarwal understanding icse mathematics class 10 Solutions for Chapter 12 similarity

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

It is given that ∆DEF ~ ∆RPQ. Is it true to say that ∠D = ∠R and ∠F = ∠P ? Why?

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.

In the figure (1) given below, AP = 2PB and CP = 2PD.

Prove that ∆ACP is similar to ∆BDP and AC || BD.

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

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

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.

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):

If in two right triangles, one of the acute angle of one triangle is equal to an acute angle of the other triangle, can you say that the two triangles are similar? Why?

In the figure given below, ∠P = ∠RTS.

Prove that ∆RPQ ~ ∆RTS.

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

Prove that ∆ACO ~ ∆BDO.

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

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

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 x MP.

In the figure (ii) given below, ∠ADC = ∠BAC. Prove that CA² = DC x BC

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.

In the figure (1) given below, AP = 2PB and CP = 2PD.

If AC = 4.5 cm, calculate the length of BD.

In the figure (2) given below,

∠ADE = ∠ACB.

Prove that ∆s ABC and AED are similar.

In the figure (2) given below,

∠ADE = ∠ACB.

If AE = 3 cm, BD = 1 cm and AB = 6 cm, calculate AC.

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.

In the given figure, DB ⊥ BC, DE ⊥ AB and AC ⊥ BC. Prove that BE/DE = AC/BC

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

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):

In ∆ABC, ∠A is acute. BD and CE are perpendicular on AC and AB respectively. Prove that AB x AE = AC x AD.

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.

Prove that triangle RLQ is similar to triangle PLN. Hence, find PN.

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.

The altitude BN and CM of ∆ABC meet at H.

Prove that CN × HM = BM × HN

The altitude BN and CM of ∆ABC meet at H.

Prove that ∆MHN ~ ∆BHC

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

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.

Name a triangle similar to triangle RLM. Evaluate RM.

In the given figure, CM and RN are respectively the medians of ∆ABC and ∆PQR. If ∆ABC ~ ∆PQR, prove that:CM/RN = AB/PQ

The altitude BN and CM of ∆ABC meet at H.

Prove that HC/HB = √[(CN × HN)/(BM × HM)]

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

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 EF.

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.

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

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

Prove that AG : GD = 2 : 1

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 AC.

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

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.

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?

In the figure (i) given below if DE || BG, AD = 3 cm, BD = 4 cm and BC = 5 cm.

Find AE : EC

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.

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

Find CQ and BQ.

In the given figure, DE || BC.

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

In the given figure, DE || BC.

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

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: PE = 3.9 cm, EQ = 3 cm, PF = 8 cm and RF = 9 cm.

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: PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm and PF = 0.36 cm.

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.

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

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

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.

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

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

Calculate LM : QR, given that BM : MC = 1 : 2.

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

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

Prove that BM/MC = AL/LQ

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

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

In the figure (i) given below if DE || BG, AD = 3 cm, BD = 4 cm and BC = 5 cm.

Find DE.

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.

Given that ∆s ABC and PQR are similar.

Find: The ratio of the area of ∆ABC to the area of ∆PQR if their corresponding sides are in the ratio 1 : 3.

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

∆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.

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

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.

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

Find CD and OB.

In the given figure, DE || BC.

If area of ∆ABC = 18cm2, find the area of trapezium DBCE

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

Find the ratio of areas of ∆OAB and ∆OCD.

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.

In the given figure, DE || BC.

Given that AD = ½ BD, calculate DE if BC = 4.5 cm.

In the given figure, DE || BC.

Prove that ∆ADE and ∆ABC are similar.

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

If AB = 6 cm: DE = 4 cm and AC = 15 cm, calculate CD.

In the figure (ii) given below, ABCD is a parallelogram. AM ⊥ DC and AN ⊥ CB. If AM = 6 cm, AN = 10 cm and the area of parallelogram ABCD is 45 cm², find BC.

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

Prove that ∆ABC ~ ∆DEC

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

Prove that ∆DEF is similar to ∆CBF. Hence, find EF/FB.

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

Find the ratio of the area of ∆ABC : area of ∆DEC.

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

Determine the ratios AD/AB, DE/BC

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

What is the ratio of the areas of ∆DEF and ∆CBF?

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

Prove that ∆OAB ~ ∆OCD.

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

Find: area ∆APO : area ∆ABC.

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.

In the figure (ii) given below, ABCD is a parallelogram. AM ⊥ DC and AN ⊥ CB. If AM = 6 cm, AN = 10 cm and the area of parallelogram ABCD is 45 cm², find AB.

In the figure (iii) given below, ABCD is a parallelogram. E is a point on AB, CE intersects the diagonal BD at O and EF || BC. If AE : EB = 2 : 3, find EF : AD.

In the figure (ii) given below, ABCD is a parallelogram. AM ⊥ DC and AN ⊥ CB. If AM = 6 cm, AN = 10 cm and the area of parallelogram ABCD is 45 cm², find area of ∆ADM : area of ∆ANB.

In the figure (iii) given below, ABCD is a parallelogram. E is a point on AB, CE intersects the diagonal BD at O and EF || BC. If AE : EB = 2 : 3, find area of ∆BEF : area of ∆ABD.

Given that ∆s ABC and PQR are similar.

Find: the ratio of their corresponding sides if area of ∆ABC : area of ∆PQR = 25 : 36.

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 area of parallelogram ABCD.

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 area of ∆BPQ.

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 area ∆CDP.

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 BE.

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.

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 ED.

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

Find, area of ∆ADE : area of quadrilateral BCED.

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 area of ∆EDC : area of trapezium ABCD.

In the figure given below, ABCD is a trapezium in which DC is parallel to AB. If AB = 9 cm, DC = 6 cm and BD = 12 cm., find BP.

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.

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

Prove that: ∆ADE ~ ∆ACB.

In the figure given below, ∠ABC = ∠DAC and AB = 8 cm, AC = 4 cm, AD = 5 cm. Find BC and CD.

In the figure given below, ∠ABC = ∠DAC and AB = 8 cm, AC = 4 cm, AD = 5 cm. Find the area of ∆ACD : area of ∆ABC.

In the figure (iii) given below, ABCD is a parallelogram. E is a point on AB, CE intersects the diagonal BD at O and EF || BC. If AE : EB = 2 : 3, find area of ∆ABD : area of trapezium AFED.

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

Prove that: If AC = 13 cm, BC = 5 cm and AE = 4 cm. Find DE and AD.

Two isosceles triangles have equal vertical angles and their areas are in the ratio 7: 16.

Find the ratio of their corresponding height.

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 the actual length of AB in km. 

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 the area of the plot in sq. km:

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: the distance of a diagonal of the plot in km. 

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:  the area of the plot in sq. km.

The model of a building is constructed with the scale factor 1 : 30. 

If the height of the model is 80 cm, find the actual height of the building in metres. 

The model of a building is constructed with the scale factor 1 : 30. 

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.

A model of a ship is made to a scale of 1 : 200. 

If the length of the model is 4 m, find the length of the ship. 

A model of a ship is made to a scale of 1 : 200. 

If the area of the deck of the ship is 160000 m², find the area of the deck of the model. 

A model of a ship is made to a scale of 1 : 200. 

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

In the figure (iii) given below, ABCD is a parallelogram. E is a point on AB, CE intersects the diagonal BD at O and EF || BC. If AE : EB = 2 : 3, find area of ∆FEO : area of ∆OBC.

In the figure given below, ABCD is a trapezium in which DC is parallel to AB. If AB = 9 cm, DC = 6 cm and BD= 12 cm., find the ratio of areas of ∆APB and ∆DPC.