Nootan solutions for मैथमैटिक्स [अंग्रेजी] कक्षा १० आईसीएसई chapter 13 - Similarity [Latest edition]

State which pair of triangles in the figure given below is similar. Write the similarity criterion used, and also write the pair of similar triangles in symbolic form.

State which pair of triangles in the figure given below is similar. Write the similarity criterion used, and also write the pair of similar triangles in symbolic form.

In the following figure, AB || DC, AB = 6 cm, AE = 3 cm, CE = 4 cm, ED = 8 cm, determine BE and CD.

If ΔАВС ~ ΔEDF and given that AB = 5 cm, BC = 6.25 cm, DF = 15 cm, and EF = 16.8 cm, find AC and DE.

If ΔАВС ~ ΔDEF, DE = 6 cm, EF = 9 cm, FD = 12 cm and AB = 3 cm, then find the perimeter of ΔAВС.

In the following figure, DE || BC and `(AD)/(DB) = 3/4`. If AC = 8.4 cm, find EC.

In the following figure, AB = 9 cm, CD = 6 cm and CM = 4 cm. Find AM.

In the following figure, DE || BC, AD = 4 cm, AB = 10 cm, BC = 12 cm, and AE = 6 cm, find AC and DE.

In the following figure, ∠ ACB = ∠ CDA. If AD = 6 cm and BD = 18 cm, find AC.

In the following figure, ABCD is a trapezium in which AB || DC. The diagonals AC and DB intersect at O.

1. Prove that ΔOAB ∼ ΔОСD
2. If OA = 3x − 1, OB = 2x + 1, OC = x + 3 and OD = 5x + 1, find the value of x.

In the following figure, ∠ACB = ∠AMN. Prove that ΔAMN ∼ ΔАСВ.

In the following figure, ∠BAD = ∠ACD. Prove that DA² = CD × BD.

The diagonals of trapezium PQRS intersect each other at O. Prove that: `(PO)/(OR) = (QO)/(OS)`

In the following figure, DE || AC and `(BE)/(EC) = (BC)/(CF)`, prove that DC || AF.

In the following figure, ABCD is a trapezium in which AB || DC. E and F are points on AD and BC, respectively, such that EF || CD. Prove that `(AE)/(ED) = (BF)/(FC)`

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

In the given figure, ∠PQR = ∠PRQ and QR × PR = QT × SQ. Show that ΔPQS ~ ΔTOR.

In the following figure, ∠ABD = ∠CDB = ∠EFD = 90°. If AB = x, CD = y, EF = z, prove that `1/z = 1/x + 1/y`:

In the following figure, AB || EF || CD. If AB = 7.5 cm, CD = 4.5 cm, EC = 3 cm, EF = x and BE = y, find x and y.

A 15-meter-high tower casts a shadow of 24 meters long at a certain time. At the same time, a pole casts a shadow 40 meters long. Find the height of the pole.

In the following figure, BQ and CP are the altitudes on AC and AB, respectively. Prove that:

1. BP × OQ = CQ × OP
2. ΔРОQ − ΔВОС

In the following figure, DE || BC and DC || FE. Prove that AD² = AB × AF.

In the following figure: DE || AQ and DF || AR. Prove that EF || QR.

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 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 the following figure, medians AD and BE of ΔABC meet at point G and DF || BE. Prove that: EF = FC

In the following figure, medians AD and BE of ΔABC meet at point G and DF || BE. Prove that: AG : GD = 2 : 1

In the following figure, AB = AC and AB² = BD × CE. Prove that ΔADB ~ ΔЕАС.

In the following figure, AD is the median of ΔABC. The bisectors of ∠ADB and ∠ADC meet AB and AC in points E and F, respectively. Prove that: EF || ВС.

In the following figure, A, B and C are points on OP, OQ and OR, respectively, such that AB || PQ and AC || PR. Show that BC || QR.

ABCD is a quadrilateral in which AB = AD, the bisector of ∠BAC and ∠CAD intersect the sides BC and CD at the points E and F, respectively. Prove that EF || BD.

If ΔABC and ΔDEF are similar and AB : DE = 5 : 4, find the ratio of the area of ΔАBC and the area of ΔDEF.

The ratio of the areas of two similar triangles is 49 : 100. Find the ratio of the lengths of their corresponding sides.

The area of ΔABC is 9 cm², and the area of ΔDEF = 16 cm². If ΔABC ~ ΔDEF, find: AB : DE.

The area of ΔABC is 9 cm², and the area of ΔDEF = 16 cm². If ΔABC ~ ΔDEF, find: perimeter (ΔABC) : perimeter (ΔABC).

The ratio between the areas of two similar triangles is 64 : 49. Find the ratio between their corresponding medians.

The ratio between the areas of two similar triangles is 64 : 49. Find the ratio between their corresponding altitudes.

ΔABC and ΔEDB are two similar triangles such that BD = `1/2 BC`. If ar(ΔАВC) = 400 cm², find the area (ΔEDB).

In ΔPQR, MN is parallel to QR and `(PM)/(MQ) = 2/3`

1. Find `(MN)/(QR)`.
2. Prove that ΔOMN and ΔORQ are similar.
3. Find the area of ΔOMN : Area of ΔORQ.

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. Prove ΔPQR ∼ ΔSPR.
2. Find the lengths of QR and PS.
3. Find `(Area  of ΔPQR)/(Area  of ΔSPR)`

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 the area of ΔACD : area of ΔABC.

D, E and F are respectively the mid-points of sides AB, BC and CA of ΔABC. Find the ratio of the area of ΔDEF and ΔABC.

In the given figure, ABC and CEF are two triangles where BA is parallel to CE and AF : AC = 5 : 8.

1. Prove that ΔADF ∼ ΔCEF.
2. Find AD if CE = 6 cm.
3. If DF is parallel to BC, find the area of ΔADF : area of ΔABC.

In the given diagram, ΔADB and ΔACB are two right-angled triangles with ∠ADB = ∠BCA = 90°. If AB = 10 cm, AD = 6 cm, BC = 2.4 cm and DP = 4.5 cm.

1. Prove that ΔAPD ∼ ΔBPC
2. Find the length of BD and PB
3. Hence, find the length of PA
4. Find area ΔAPD : area ΔBPC.

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

1. If the height of the model is 80 cm, find the actual height of the building in metres.
2. 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.

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:

1. The actual length of AB in km.
2. The area of the plot in sq. km.

On a map drawn to a scale of 1 : 25000, a rectangular plot of land, ABCD, is measured as AB = 12 cm and BC = 16 cm. Calculate the diagonal distance of the plot in km and the plot area in km². 

A model of a high-rise building is made to a scale of 1 : 50.

1. If the height of the model is 0.8 m, find the height of the actual building.
2. If the floor area of a flat in the building is 20 m², find the floor area of that in the model.

The scale of a map is 1 : 200000. A plot of land of area 20km² is to be represented on the map. Find: The number of kilometres on the ground represented by 1 cm on the map.

The scale of a map is 1 : 200000. A plot of land of area 20km² is to be represented on the map. Find: The area in km² that can be represented by 1 cm² 

In the given figure ∠BAP = ∠DCP = 70°, PC = 6 cm and CA = 4 cm, then PD : DB is ______.

In the given figure, AP = `1/2 PB`, then ar(ΔAPQ) : ar(ΔABC) is ______.

If ΔABC ~ ΔDEF and ar(ΔABC) : ar(ΔDEF) = 25 : 4, then BC : EF is ______.

ΔАВС ~ ΔРQR, ar(ΔABC) = 16 cm², ar(ΔPQR) = 144 cm² and PQ = 6 cm, then the length of BA is ______.

In the given figure, DE || BC and `(AD)/(DB) = 2/5`. If AC = 5.6 cm, then EC is equal to ______.

In the given figure, ΔАВС ~ ΔRPQ, then ∠Q is equal to ______.

In the given figure, AE || DC. If AB = 6 cm, BC = 4 cm, and AE = 9 cm, then DC is equal to ______.

If ΔАBC ~ ΔPQR, PQ = 5 cm, perimeter of ΔPQR = 20 cm, AB = 7.5 cm, then perimeter of ABC is ______.

∆ABC and ∆BDE are two equilateral triangles such that D is the mid-point of BC. The ratio of the areas of triangles ABC and BDE is ______.

The areas of two similar triangles are 121 cm² and 100 cm². If the altitude of a larger triangle is 6.6 cm, then the corresponding altitude of the smaller triangle is ______.

In ΔABC and ΔPQR, ∠A = ∠P, ∠B = ∠Q and AB = 2 PQ, then the two triangles are ______.

Which of the following is not a criterion of similarity of triangles?

In the given diagram, ΔABC ∼ ΔPQR. If AD and PS are the bisectors of ∠BAC and ∠QPR, respectively, then ______.

The diagonals of quadrilateral ABCD intersect at O. Prove that `"A(∆ACB)"/"A(∆ACD)" = "BO"/"DO"`.

If two sides and a median bisecting the third side of a triangle are respectively proportional to the corresponding sides and the median of another triangle, then prove that the two triangles are similar.

In the given figure,  ∠PQR = ∠PST = 90°, PQ = 5cm and PS =  2cm.

1. Prove that ΔPQR ~ ΔPST.
2. Find the Area of ΔPQR : Area of quadrilateral SRQT.

If the areas of two similar triangles are equal, prove that they are congruent.

Prove that the diagonals of a trapezium divide each other proportionally.

A model of a ship is made to a scale of 1 : 250. Calculate:

1. the length of the ship, if the length of the model is 1.6 m.
2. the area of the deck of the ship, if the area of the deck of the model is 2.4 m².
3. the volume of the model, if the volume of the ship is 1 km³.

The volume of a machine is 27000 cm³. A model of the machine is made, the reduction factor being 2 : 15. Find the volume of the model.

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 ΔCPQ = 20 cm², find:

1. area of ΔBPQ.
2. area of ΔCDP.

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

1. BP
2. the ratio of the areas of ΔAPB and ΔDPC

In the adjoining figure, M is the mid-point of AB, ∠A = ∠B = 90° = ∠CMD, prove that:

1. ΔDAM is similar to ΔМВС
2. `"area of ΔDAM"/"area of ΔMBC" = (AD)/(BC)`
3. `(AD)/(BC) = (MD^2)/(MC^2)`