Frank solutions for Mathematics [English] Class 9 ICSE chapter 16 - Similarity [Latest edition]

In ΔABC, D and E are the mid-point on AB and AC such that DE || BC.
If AD = 4, AE = 8, DB = x - 4 and EC = 3x - 19, find x.

In ΔABC, D and E are the mid-point on AB and AC such that DE || BC.
If AD : BD = 4 : 5 and EC = 2.5cm, find AE.

In ΔABC, D and E are the mid-point on AB and AC such that DE || BC.
If AD = 4x - 3, AE = 8x - 7, BD = 3x - 1 and CE = 5x - 3,Find x.

In ΔABC, D and E are the mid-point on AB and AC such that DE || BC.
If AD = 8cm, AB = 12cm and AE = 12cm, find CE.

If ΔABC, D and E are points on AB and AC. Show that DE || BC for each of the following case or not:
AB = 5.6cm, AD = 1.4cm, AC = 7.2cm, and AE = 1.8cm

If ΔABC, D and E are points on AB and AC. Show that DE || BC for each of the following case or not:
AB = 10.8cm, BD = 4.5cm, AC = 4.8cm, and AE = 2.8cm

If ΔABC, D and E are points on AB and AC. Show that DE || BC for each of the following case or not:
AD = 5.7cm, BD = 9.5cm, AE = 3.3cm, and EC = 5.5cm

In figure, PQ is parallel to BC, AP : AB = 2 : 7. If QC = 0 and BC = 21,

Find
(i) AQ
(ii) PQ

In ΔABC, DE is parallel to BC and DE = 3:8.

Find:
(i) AD : BD
(ii) AE, if AC = 16.

In ΔABC, point D divides AB in the ratio 5:7, Find: `"AE"/"EC"`

In ΔABC, point D divides AB in the ratio 5:7, Find: `"AD"/"AB"`

In ΔABC, point D divides AB in the ratio 5:7, Find: `"AE"/"AC"`

In ΔABC, point D divides AB in the ratio 5:7, Find: BC, If DE = 2.5cm

In ΔABC, point D divides AB in the ratio 5:7, Find: DE, If BC = 4.8cm

If ΔPQR, AB is drawn parallel to QR. If PQ = 9cm, PR = 6cm and PB = 4.cm, find the length of AP.

In ΔABC, MN is drawn parallel to BC. If AB = 3.5cm, AM : AB = 5 : 7 and NC = 2cm, find:
(i) AM
(ii) AC

The sides PQ and PR of the ΔPQR are produced to S and T respectively. ST is drawn parallel to QR and PQ: PS = 3:4. If PT = 9.6 cm, find PR. If 'p' be the length of the perpendicular from P to QR, find the length of the perpendicular from P to ST in terms of 'p'.

ΔABC is right angled at A. AD is drawn perpendicular to BC. If AB = 8cm and AC = 6cm, calculate BD.

In the figure, PR || SQ. If PR = 10cm, PT = 5cm, TQ = 6cm and ST = 9cm, calculate RT and SQ.

ABCD is a parallelogram whose sides AB and BC are 18cm and 12cm respectively. G is a point on AC such that CG : GA = 3 : 5 BG is produced to meet CD at Q and AD produced at P. Prove that ΔCGB ∼ ΔAGP. Hence, fi AP.

In ΔABC, BP and CQ are altitudes from B and C on AC and AB respectively. BP and CQ intersect at O. Prove that
(i) PC x OQ = QB x OP

(ii) `"OC"^2/"OB"^2 = ("PC" xx "PO")/("QB" xx "QO")`

In the figure, PQR is a straight line and PS || RT. If QS = 12cm, QR = 15cm, QT = 10cm and RT = 6cm, find PQ and PS.

The diagonal AC of a parallelogram ABCD intersects DP at the point Q, where P is any point on side AB. Prove that CQ x PQ = QA x QD.

AM and DN are the altitudes of two similar triangles ABC and DEF. Prove that: AM : DN = AB : DE.

Prove that the external bisector of an angle of a triangle divides the opposite side externally n the ratio of the sides containing the angle.

In the figure, AB || RQ and BC || SQ, prove that `"PC"/"PS" = "PA"/"PR"`.

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

PQ is perpendicular to BA and BD is perpendicular to AP.PQ and BD intersect at R. Prove that ΔABD ∼ ΔAPQ and `"AB"/"AP" = "BD"/"PQ"`.

Through the vertex S of a parallelogram PQRS, a line is drawn to intersect the sides Qp and QR produced at M and N respectively. Prove that `"SP"/"PM" = "MQ"/"QN" = "MR"/"SR"`

In a quadrilateral PQRS, the diagonals PR and QS intersect each other at the point T. If PT:TR = QT :TS = 1:2, show that ΔPTQ - DRTS

In a quadrilateral PQRS, the diagonals PR and QS intersect each other at the point T. If PT:TR = QT :TS = 1:2, show that TP:TQ = TR:TS

In the given figure, PB is the bisector of ABC and ABC =ACB. Prove that:
a. BC x AP = PC x AB
b. AB:AC = BP: BC

In a right-angled triangle ABC, ∠B = 90°, P and Q are the points on the sides AB and AC such as PQBC, AB = 8 cm, AQ = 6 cm and PA:AB = 1:3. Find the lengths of AC and BC.

Given that ΔABC ∼ ΔPRQ, name the corresponding angles and the corresponding sides.

In ΔABC, DE || BC such that AD =1.5 cm, DB = 3 cm and AE = 1 cm. Find AC.

Given is a triangle with sides 3 cm, 5 cm and 6 cm. Find the sides of a triangle which is similar to the given triangle and its shortest side is 4.5 cm.

Two figures are similar. If the ratio of their perimeters is 8:16. What will be the ratio of the corresponding sides?

Harmeet is 6 feet tall and casts a shadow of 3 feet long. What is the height of a nearby pole if it casts a shadow of 12 feet long at the same time?

The areas of two similar triangles are 16cm² and 9cm² respectively. If the altitude of the smaller triangle is 1.8cm, find the length of the altitude corresponding to the larger triangle.

The areas of two similar triangles are 169cm² and 121cm² respectively. If one side of the larger triangle is 26cm, find the length of the corresponding side of the smaller triangle.

D and E are points on the sides AB and AC  of ΔABC such that DE | | BC and divides ΔABC into two parts, equal in area. Find `"BD"/"AB"`.

In ΔABC, DE is drawn parallel to BC cutting AB in the ratio 2 : 3. Calculate:
(i) `("area"(Δ"ADE"))/("area"(Δ"ABC")`

(i) `("area"("trapeziumEDBC"))/("area"(Δ"ABC"))`

In ΔABC, AB = 8cm, AC = 10cm and ∠B = 90°. P and Q are the points on the sides AB and AC respectively such that PQ = 3cm ad ∠PQA = 90. Find: The area of ΔAQP.

In ΔABC, AB = 8cm, AC = 10cm and ∠B = 90°. P and Q are the points on the sides AB and AC respectively such that PQ = 3cm ad ∠PQA = 90. Find: Area of quadrilateral PBCQ: area of ΔABC.

Find the scale factor in each of the following and state the type of size transformation:
Image length = 6cm, Actual length = 4cm.

Find the scale factor in each of the following and state the type of size transformation:
Actual length = 12cm, Image length = 15cm.

Find the scale factor in each of the following and state the type of size transformation:
Image length = 8cm, Actual length = 20cm.

Find the scale factor in each of the following and state the type of size transformation:
Actual area = 64m², Model area = 100cm² 

Find the scale factor in each of the following and state the type of size transformation:
Model area = 75cm², Actual area = 3cm²

Find the scale factor in each of the following and state the type of size transformation:
Model volume = 200cm³, Actual volume = 8cm³

ΔABC has been reduced by a scale factor 0.6 to ΔA'B'C'/ Calculate:Length of B' C', if BC = 8cm

ΔABC has been reduced by a scale factor 0.6 to ΔA'B'C'/ Calculate: Length of AB, if A'B' = 5.4cm

ΔABC is enlarged, with a scale factor 5. Find: A'B', if AB = 4cm

ΔABC is enlarged, with a scale factor 5. Find: BC, f B'C' = 16cm

ΔXYZ is enlarged to ΔX'Y'Z'. If XY = 12cm, YZ = 8cm and XZ = 14cm and the smallest side of ΔX'Y'Z' is 12cm, find the scale factor and use it to find the length of the other sides of the image ΔX'Y'Z'.

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, ∠ABC = 90°. Calculate:
(i) The actual length of AB in km.
(ii) The area of Plot in sq. km.

The dimensions of the model of a building are 1.2m x 75cm x 2m. If the scale factor is 1 : 20; find the actual dimensions of the building.

The scale of a map is 1 : 50000. The area of a city is 40 sq km which is to be represented on the map. Find: The area of land represented on the map.

The scale of a map is 1 : 50000. The area of a city is 40 sq km which is to be represented on the map. Find: The length of a scale in km represented by 1cm on the map.

A plot of land of area 20km² is represented on the map with a scale factor of 1:200000. Find: The number of KM represented by 2cm on the map.

A plot of land of area 20km² is represented on the map with a scale factor of 1:200000. Find: The ground area in km² that is represented by 2cm² on the map.

A plot of land of area 20km² is represented on the map with a scale factor of 1:200000. Find: The area on the map that represented the plot of land.

A map is drawn to scale of 1:20000. Find: The distance covered by 6cm on the map

A map is drawn to scale of 1:20000. Find: The distance on the map representing 4km

A map is drawn to scale of 1:20000. Find: The area of the lake on the map which has an actual area of 12km² 

A model of cargo tuck is made to a scale of 1:40. The length of the model is 15cm. Calculate: The length of the truck

A model of cargo tuck is made to a scale of 1:40. The length of the model is 15cm. Calculate: The volume of the model if the volume of the truck is 6m³

A model of cargo tuck is made to a scale of 1:40. The length of the model is 15cm. Calculate: The base area of the truck, if the base area of the model is 30m² 

A model of a ship is made to a scale of 1:500. Find: The length of the ship, if length of the model is 1.2.

A model of a ship is made to a scale of 1:500. Find: The area other deck o the ship, if the area of the deck of its model is m²

A model of a ship is made to a scale of 1:500. Find: The volume of the model when the volume of the ship is 1km³ 

On a map drawn to a scale of 1:25000, a rectangular plot of land has sides 12cm x 16cm. Calculate: The diagonal distance of the plot in km

On a map drawn to a scale of 1:25000, a rectangular plot of land has sides 12cm x 16cm. Calculate: The area of the plot in sq km

On a map drawn to a scale of 1:25000, a triangular plot of land is right angled and the sides forming the right angle measure 225cm and 64cm.Find: The actual length of the sides in km

On a map drawn to a scale of 1:25000, a triangular plot of land is right angled and the sides forming the right angle measure 225cm and 64cm.Find: The area of the plot in sq. km.

In a triangle ABC, AB = 4 cm, BC = 4.5 cm and CA = 5 cm. Construct ΔABC. Find the image A'B'C of the ΔABC obtained by enlarging it by a scale factor 2. Measure the sides of the image A'B'C' and show that AB:A'B' = AC:B'C' = CA:C'A'