Selina solutions for Mathematics [English] Class 10 ICSE chapter 15 - Similarity (With Applications to Maps and Models) [Latest edition]

In the figure, given below, straight lines AB and CD intersect at P; and AC || BD. Prove that: ΔAPC and ΔBPD are similar.

In the figure, given below, straight lines AB and CD intersect at P; and AC || BD. Prove that: If BD = 2.4 cm, AC = 3.6 cm, PD = 4.0 cm and PB = 3.2 cm; find the lengths of PA and PC.

In a trapezium ABCD, side AB is parallel to side DC; and the diagonals AC and BD intersect each other at point P. Prove that :

1. ΔAPB is similar to ΔCPD.
2. PA × PD = PB × PC. 

In a trapezium ABCD, side AB is parallel to side DC; and the diagonals AC and BD intersect each other at point P. Prove that : PA x PD = PB x PC. 

P is a point on side BC of a parallelogram ABCD. If DP produced meets AB produced at point L, prove that: DP : PL = DC : BL.

P is a point on side BC of a parallelogram ABCD. If DP produced meets AB produced at point L, prove that: DL : DP = AL : DC.

In quadrilateral ABCD, the diagonals AC and BD intersect each other at point O. If AO = 2CO and BO = 2DO; show that: ΔAOB is similar to ΔCOD.

In quadrilateral ABCD, the diagonals AC and BD intersect each other at point O. If AO = 2CO and BO = 2DO; show that: OA × OD = OB × OC.

In ΔABC, angle ABC is equal to twice the angle ACB, and bisector of angle ABC meets the opposite side at point P. Show that: CB : BA = CP : PA

In ΔABC, angle ABC is equal to twice the angle ACB, and bisector of angle ABC meets the opposite side at point P. Show that: AB × BC = BP × CA 

In ΔABC; BM ⊥ AC and CN ⊥ AB; show that:

`(AB)/(AC) = (BM)/(CN) = (AM)/(AN)`

In the given figure, DE || BC, AE = 15 cm, EC = 9 cm, NC = 6 cm and BN = 24 cm.

1. Write all possible pairs of similar triangles.
2. Find lengths of ME and DM.

In the given figure, DE || BC, AE = 15 cm, EC = 9 cm, NC = 6 cm and BN = 24 cm. Find lengths of ME and DM.

In the given figure, AD = AE and AD² = BD × EC. Prove that: triangles ABD and CAE are similar. 

In the given figure, AB || DC, BO = 6 cm and DQ = 8 cm; find: BP × DO.

Angle BAC of triangle ABC is obtuse and AB = AC. P is a point in BC such that PC = 12 cm. PQ and PR are perpendiculars to sides AB and AC respectively. If PQ = 15 cm and PR = 9 cm; find the length of PB.

State, true or false:

Two similar polygons are necessarily congruent.

State, true or false:

Two congruent polygons are necessarily similar.

State, true or false:

All equiangular triangles are similar.

State, true or false:

All isosceles triangles are similar.

State, true or false:

Two isosceles-right triangles are similar.

State, true or false:

Two isosceles triangles are similar, if an angle of one is congruent to the corresponding angle of the other.

State, true or false:

The diagonals of a trapezium divide each other into proportional segments.

Given: ∠GHE = ∠DFE = 90°,

DH = 8, DF = 12,

DG = 3x – 1 and DE = 4x + 2.

Find: the lengths of segments DG and DE.

D is a point on the side BC of ∆ABC such that ∠ADC = ∠BAC. Prove that` \frac{"CA"}{"CD"}=\frac{"CB"}{"CA"} or "CA"^2 = "CB" × "CD".`

In the given figure, ∆ABC and ∆AMP are right angled at B and M respectively.

Given AC = 10 cm, AP = 15 cm and PM = 12 cm.

1. Prove that: ∆ABC ~ ∆AMP
2. Find: AB and BC.

Given: RS and PT are altitudes of ΔPQR. Prove that:

1. ΔPQT ~ ΔQRS,
2. PQ × QS = RQ × QT.

Given: ABCD is a rhombus, DPR and CBR are straight lines.

Prove that: DP × CR = DC × PR.

Given: FB = FD, AE ⊥ FD and FC ⊥ AD.

Prove that: `(FB)/(AD) = (BC)/(ED)`.

In ∆PQR, ∠Q = 90° and QM is perpendicular to PR. Prove that:

1. PQ² = PM × PR
2. QR² = PR × MR
3. PQ² + QR² = PR²

In ∆ABC, ∠B = 90° and BD ⊥ AC.

1. If CD = 10 cm and BD = 8 cm; find AD.
2. If AC = 18 cm and AD = 6 cm; find BD.
3. If AC = 9 cm and AB = 7 cm; find AD.

In the figure, PQRS is a parallelogram with PQ = 16 cm and 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.

Find the lengths of PN and RM.

In quadrilateral ABCD, diagonals AC and BD intersect at point E such that
AE : EC = BE : ED. Show that: ABCD is a trapezium.

In triangle ABC, AD is perpendicular to side BC and AD² = BD × DC. Show that angle BAC = 90°.

In the given figure, AB || EF || DC; AB = 67.5 cm, DC = 40.5 cm and AE = 52.5 cm.

1. Name the three pairs of similar triangles.
2. Find the lengths of EC and EF.

In the given figure, QR is parallel to AB and DR is parallel to AB and DR is parallel to QB.

Prove that: PQ² = PD × PA.

Through the mid-point M of the side CD of a parallelogram ABCD, the line BM is drawn intersecting diagonal AC in L and AD produced in E. Prove that: EL = 2BL.

In the given figure, P is a point on AB such that AP : PB = 4 : 3. PQ is parallel to AC.

1. Calculate the ratio PQ : AC, giving reason for your answer.
2. In triangle ARC, ∠ARC = 90° and in triangle PQS, ∠PSQ = 90°. Given QS = 6 cm, calculate the length of AR.

In the right-angled triangle QPR, PM is an altitude.

Given that QR = 8 cm and MQ = 3.5 cm, calculate the value of PR.

In the figure, given below, the medians BD and CE of a triangle ABC meet at G. Prove that:

1. ΔEGD ~ ΔCGB and
2. BG = 2GD from (i) above.

In the following figure, point D divides AB in the ratio 3 : 5. Find :

1. `(AE)/(EC)`
2. `(AD)/(AB)`
3. `(AE)/(AC)`
   Also, if:
4. DE = 2.4 cm, find the length of BC.
5. BC = 4.8 cm, find the length of DE.

In the following figure, point D divides AB in the ratio 3 : 5. Find : `(AD)/(AB)`

In the following figure, point D divides AB in the ratio 3 : 5. Find : `(AE)/(AC)`

In the following figure, point D divides AB in the ratio 3 : 5. Find :

DE = 2.4 cm, find the length of BC.

In the following figure, point D divides AB in the ratio 3 : 5. Find :

BC = 4.8 cm, find the length of DE.

In the given figure, PQ || AB; CQ = 4.8 cm QB = 3.6 cm and AB = 6.3 cm. Find :

1. `(CP)/(PA)` 
2. PQ
3. If AP = x, then the value of AC in terms of x.

In the given figure, PQ || AB; CQ = 4.8 cm QB = 3.6 cm and AB = 6.3 cm. Find : PQ

In the given figure, PQ || AB; CQ = 4.8 cm QB = 3.6 cm and AB = 6.3 cm. Find : If AP = x, then the value of AC in terms of x.

A line PQ is drawn parallel to the side BC of ΔABC which cuts side AB at P and side AC at Q. If AB = 9.0 cm, CA = 6.0 cm and AQ = 4.2 cm, find the length of AP.

In ΔABC, D and E are the points on sides AB and AC respectively. Find whether DE || BC, if:

1. AB = 9 cm, AD = 4 cm, AE = 6 cm and EC = 7.5 cm.
2. AB = 6.3 cm, EC = 11.0 cm, AD = 0.8 cm and AE = 1.6 cm.

In ΔABC, D and E are the points on sides AB and AC respectively. Find whether DE || BC, if AB = 6.3 cm, EC = 11.0 cm, AD = 0.8 cm and AE = 1.6 cm.

In the given figure, ΔABC ~ ΔADE. If AE : EC = 4 : 7 and DE = 6.6 cm, find BC. If 'x' be the length of the perpendicular from A to DE, find the length of perpendicular from A to BC in terms of 'x'. 

A line segment DE is drawn parallel to base BC of ΔABC which cuts AB at point D and AC at point E. If AB = 5BD and EC = 3.2 cm, find the length of AE.

In the figure, given below, AB, CD and EF are parallel lines. Given AB = 7.5 cm, DC = y cm, EF = 4.5 cm, BC = x cm and CE = 3 cm, calculate the values of x and y.

In the figure, given below, PQR is a right-angle triangle right angled at Q. XY is parallel to QR, PQ = 6 cm, PY = 4 cm and PX : XQ = 1 : 2. Calculate the lengths of PR and QR.

In the following figure, M is mid-point of BC of a parallelogram ABCD. DM intersects the diagonal AC at P and AB produced at E. Prove that : PE = 2 PD 

The given figure shows a parallelogram ABCD. E is a point in AD and CE produced meets BA produced at point F. If AE = 4 cm, AF = 8 cm and AB = 12 cm, find the perimeter of the parallelogram ABCD.

The ratio between the corresponding sides of two similar triangles is 2 is to 5. Find the ratio between the areas of these triangles.

Area of two similar triangles are 98 sq. cm and 128 sq. cm. Find the ratio between the lengths of their corresponding sides.

A line PQ is drawn parallel to the base BC of ∆ABC which meets sides AB and AC at points P and Q respectively. If AP = `1/3` PB; find the value of:

1. `"Area of ΔABC"/"Area of ΔAPQ"`
2. `"Area of ΔAPQ"/"Area of trapezium PBCQ"`

The perimeter of two similar triangles are 30 cm and 24 cm. If one side of the first triangle is 12 cm, determine the corresponding side of the second triangle.

In the given figure, AX : XB = 3 : 5

Find:

1. the length of BC, if the length of XY is 18 cm.
2. the ratio between the areas of trapezium XBCY and triangle ABC.

ABC is a triangle. PQ is a line segment intersecting AB in P and AC in Q such that PQ || BC and divides triangle ABC into two parts equal in area. Find the value of ratio BP : AB.

In the given triangle PQR, LM is parallel to QR and PM : MR = 3 : 4.

Calculate the value of ratio:

1. `(PL)/(PQ)` and then `(LM)/(QR)`
2. `"Area of ΔLMN"/"Area of ΔMNR"`
3. `"Area of ΔLQM"/"Area of ΔLQN"`

The given diagram shows two isosceles triangles which are similar. In the given diagram, PQ and BC are not parallel; PC = 4, AQ = 3, QB = 12, BC = 15 and AP = PQ.

Calculate:

1. the length of AP,
2. the ratio of the areas of triangle APQ and triangle ABC.

In the figure, given below, ABCD is a parallelogram. P is a point on BC such that BP : PC = 1 : 2. DP produced meets AB produces at Q. Given the area of triangle CPQ = 20 cm².

Calculate:

1. area of triangle CDP,
2. area of parallelogram ABCD.
   

In the given figure, BC is parallel to DE. Area of triangle ABC = 25 cm², Area of trapezium BCED = 24 cm² and DE = 14 cm. Calculate the length of BC. Also, find the area of triangle BCD.

The given figure shows a trapezium in which AB is parallel to DC and diagonals AC and BD intersect at point P. If AP : CP = 3 : 5,

Find:

1. ∆APB : ∆CPB
2. ∆DPC : ∆APB
3. ∆ADP : ∆APB
4. ∆APB : ∆ADB

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

1. Determine the ratios `(AD)/(AB)` and `(DE)/(BC)`. 
2. Prove that ∆DEF is similar to ∆CBF. Hence, find `(EF)/(FB).`
3. What is the ratio of the areas of ∆DEF and ∆BFC?

In the given figure, ∠B = ∠E, ∠ACD = ∠BCE, AB = 10.4 cm and DE = 7.8 cm. Find the ratio between areas of the ∆ABC and ∆DEC.

A triangle ABC has been enlarged by scale factor m = 2.5 to the triangle A' B' C'. Calculate : the length of AB, if A' B' = 6 cm.

A triangle ABC has been enlarged by scale factor m = 2.5 to the triangle A' B' C'. Calculate : the length of C' A' if CA = 4 cm. 

A triangle LMN has been reduced by scale factor 0.8 to the triangle L' M' N'. Calculate: the length of M' N', if MN = 8 cm.

A triangle LMN has been reduced by scale factor 0.8 to the triangle L' M' N'. Calculate: the length of LM, if L' M' = 5.4 cm.

A triangle ABC is enlarged, about the point O as centre of enlargement, and the scale factor is 3. Find : A' B', if AB = 4 cm.

A triangle ABC is enlarged, about the point O as centre of enlargement, and the scale factor is 3. Find : BC, if B' C' = 15 cm. 

A triangle ABC is enlarged, about the point O as centre of enlargement, and the scale factor is 3. Find : OA, if OA' = 6 cm.

A triangle ABC is enlarged, about the point O as centre of enlargement, and the scale factor is 3. Find : OC', if OC = 21 cm.

Also, state the value of :

1. `(OB^')/(OB)`
2. `(C^'A^')/(CA)`

A model of an aeroplane is made to a scale of 1 : 400. Calculate : the length, in cm, of the model; if the length of the aeroplane is 40 m.

A model of an aeroplane is made to a scale of 1 : 400. Calculate : the length, in m, of the aeroplane, if length of its model is 16 cm.

The dimensions of the model of a multistorey building are 1.2 m × 75 cm × 2 m. If the scale factor is 1 : 30; find the actual dimensions of the building.

On a map drawn to a scale of 1 : 2,50,000; a triangular plot of land has the following measurements : AB = 3 cm, BC = 4 cm and  ∠ABC = 90°.

Calculate : the actual lengths of AB and BC in km. 

On a map drawn to a scale of 1 : 2,50,000; a triangular plot of land has the following measurements : AB = 3 cm, BC = 4 cm and angle ABC = 90°.

Calculate : the area of the plot in sq. km.

A model of a ship is made to a scale 1 : 300.

1. The length of the model of the ship is 2 m. Calculate the length of the ship.
2. The area of the deck of the ship is 180,000 m². Calculate the area of the deck of the model.
3. The volume of the model is 6.5 m³. Calculate the volume of the ship.

In the following figure, XY is parallel to BC, AX = 9 cm, XB = 4.5 cm and BC = 18 cm.

Find :

1. `(AY)/(YC)`
2. `(YC)/(AC)`
3. XY

In the following figure, XY is parallel to BC, AX = 9 cm, XB = 4.5 cm and BC = 18 cm.

Find : `(YC)/(AC)`

In the following figure, XY is parallel to BC, AX = 9 cm, XB = 4.5 cm and BC = 18 cm.

Find : XY

In the following figure, ABCD to a trapezium with AB || DC. If AB = 9 cm, DC = 18 cm, CF = 13.5 cm, AP = 6 cm and BE = 15 cm.

Calculate:

1. EC
2. AF
3. PE

In the following figure, ABCD to a trapezium with AB || DC. If AB = 9 cm, DC = 18 cm, CF = 13.5 cm, AP = 6 cm and BE = 15 cm, Calculate: AF

In the following figure, ABCD to a trapezium with AB ‖ DC. If AB = 9 cm, DC = 18 cm, CF = 13.5 cm, AP = 6 cm and BE = 15 cm, Calculate: PE

In the following figure, AB, CD and EF are perpendicular to the straight line BDF.

If AB = x and CD = z unit and EF = y unit, prove that : `1/x + 1/y = 1/z`

Triangle ABC is similar to triangle PQR. If AD and PM are corresponding medians of the two triangles, prove that : `("AB")/("PQ") = ("AD")/("PM")`.

Triangle ABC is similar to triangle PQR. If AD and PM are altitudes of the two triangles, prove that : `(AB)/(PQ) = (AD)/(PM)`.

Triangle ABC is similar to triangle PQR. If bisector of angle BAC meets BC at point D and bisector of angle QPR meets QR at point M, prove that : `(AB)/(PQ) = (AD)/(PM)`.

In the following figure, ∠AXY = ∠AYX. If `(BX)/(AX) = (CY)/(AY)`, show that triangle ABC is isosceles.

In the following diagram, lines l, m and n are parallel to each other. Two transversals p and q intersect the parallel lines at points A, B, C and P, Q, R as shown.

Prove that : `(AB)/(BC) = (PQ)/(QR)`

In the following figure, DE || AC and DC || AP. Prove that : `(BE)/(EC) = (BC)/(CP)`.

In the figure given below, AB || EF || CD. If AB = 22.5 cm, EP = 7.5 cm, PC = 15 cm and DC = 27 cm.

Calculate :

1. EF
2. AC

In the figure given below, AB ‖ EF ‖ CD. If AB = 22.5 cm, EP = 7.5 cm, PC = 15 cm and DC = 27 cm. Calculate : AC

In ΔABC, ∠ABC = ∠DAC, AB = 8 cm, AC = 4 cm and AD = 5 cm.

1. Prove that ΔACD is similar to ΔBCA.
2. Find BC and CD.
3. Find area of ΔACD : area of ΔABC.

In the given triangle P, Q and R are the mid-points of sides AB, BC and AC respectively. Prove that triangle PQR is similar to triangle ABC.

In the following figure, AD and CE are medians of ΔABC. DF is drawn parallel to CE. Prove that :

1. EF = FB,
2. AG : GD = 2 : 1

The two similar triangles are equal in area. Prove that the triangles are congruent.

The ratio between the altitudes of two similar triangles is 3 : 5; write the ratio between their :

1. corresponding medians.
2. perimeters.
3. areas.

The ratio between the areas of two similar triangles is 16 : 25. State the ratio between their :

1. perimeters.
2. corresponding altitudes.
3. corresponding medians.

The given figure shows a triangle PQR in which XY is parallel to QR. If PX : XQ = 1 : 3 and QR = 9 cm, find the length of XY.

Further, if the area of ΔPXY = x cm²; find, in terms of x the area of :

1. triangle PQR.
2. trapezium XQRY.

On a map, drawn to a scale of 1 : 20000, a rectangular plot of land ABCD has AB = 24 cm and BC = 32 cm. Calculate : 

1. the diagonal distance of the plot in kilometer.
2. the area of the plot in sq. km.

The dimensions of the model of a multistoreyed building are 1 m by 60 cm by 1.20 m. If the scale factor is 1 : 50, find the actual dimensions of the building. 

Also, find: 

1. the floor area of a room of the building, if the floor area of the corresponding room in the model is 50 sq. cm.
2. the space (volume) inside a room of the model, if the space inside the corresponding room of the building is 90 m³.

In a triangle PQR, L and M are two points on the base QR, such that ∠LPQ = ∠QRP and ∠RPM = ∠RQP. Prove that:

1. ΔPQL ∼ ΔRPM
2. QL × RM = PL × PM
3. PQ² = QR × QL

A triangle ABC with AB = 3 cm, BC = 6 cm and AC = 4 cm is enlarged to ΔDEF such that the longest side of ΔDEF = 9 cm. Find the scale factor and hence, the lengths of the other sides of ΔDEF.

Two isosceles triangles have equal vertical angles. Show that the triangles are similar. If the ratio between the areas of these two triangles is 16 : 25, find the ratio between their corresponding altitudes.

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

Find:

1. area ΔAPO : area ΔABC.
2. area ΔAPO : area ΔCQO.

The following figure shows a triangle ABC in which AD and BE are perpendiculars to BC and AC respectively. 

Show that:

1. ΔADC ∼ ΔBEC
2. CA × CE = CB × CD
3. ΔABC ~ ΔDEC
4. CD × AB = CA × DE

In the give figure, ABC is a triangle with ∠EDB = ∠ACB. Prove that ΔABC ∼ ΔEBD. If BE = 6 cm, EC = 4 cm, BD = 5 cm and area of ΔBED = 9 cm². Calculate the: 

1. length of AB
2. area of ΔABC

In the given figure, ABC is a right angled triangle with ∠BAC = 90°.

1. Prove that : ΔADB ∼ ΔCDA.
2. If BD = 18 cm and CD = 8 cm, find AD.
3. Find the ratio of the area of ΔADB is to area of ΔCDA.

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

Prove that : ΔABC ~  ΔDEC 

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

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

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

Find the ratio of the area of a ΔABC : area of ΔDEC.

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

ΔADE ~ ΔACB.

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.

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 ΔADE : area of quadrilateral BCED.

Given : AB || DE and BC || EF. Prove that :

1. `(AD)/(DG) = (CF)/(FG)`
2. ∆DFG ∼ ∆ACG

PQR is a triangle. S is a point on the side QR of ΔPQR such that ∠PSR = ∠QPR. Given QP = 8 cm, PR = 6 cm and SR = 3 cm.

1. i. ProveΔPQR∼ Δ
2. Find the lengths of QR and PS.
3. `(Area   of DeltaPQR)/(area   of Delta SPR)`