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In Each of the Given Pairs of Triangles, Find Which Pair of Triangles Are Similar. State the Similarity Criterion and Write the Similarity Relation in Symbolic Form: - Mathematics

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In each of the given pairs of triangles, find which pair of triangles are similar. State the similarity criterion and write the similarity relation in symbolic form: 

 

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In Δ ABC
∠A + ∠ЁЭР╡+ ∠ЁЭР╢=1800 (Angle Sum Property)
тЯ╣80°+ ∠ЁЭР╡+700=180°
тЯ╣ ∠ЁЭР╡=30°
∠ЁЭР┤= ∠ЁЭСА ЁЭСОЁЭСЫЁЭСС ∠ЁЭР╡= ∠ЁЭСБ
Therefore, by AA similarity , Δ ABC - Δ MNR 

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рдкрд╛рда 4: Triangles - Exercises 2

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In figure, find ∠L


E and F are points on the sides PQ and PR, respectively, of a ΔPQR. For the following case, state whether EF || QR:

PE = 4 cm, QE = 4.5 cm, PF = 8 cm and RF = 9 cm


Given: ∠GHE = ∠DFE = 90°,

DH = 8, DF = 12,

DG = 3x – 1 and DE = 4x + 2.


Find: the lengths of segments DG and DE.


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.


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. тИЖAPB : тИЖCPB
  2. тИЖDPC : тИЖAPB
  3. тИЖADP : тИЖAPB
  4. тИЖAPB : тИЖADB

In the given figure, if ∠ADE = ∠B, show that ΔADE ~ ΔABC. If AD = 3.8cm, AE = 3.6cm, BE = 2.1cm and BC = 4.2cm, find DE.  


In the given figure, ∠CAB = 90° and AD⊥BC. Show that ΔBDA ~ ΔBAC. If AC = 75cm, AB = 1m and BC = 1.25m, find AD. 

 


A vertical pole of length 7.5 cm casts a shadow 5 m long on the ground and at the same time a tower casts a shadow 24 m long. Find the height of the tower.


In an isosceles ΔABC, the base AB is produced both ways in P and Q such that
AP × BQ = AC2.
Prove that ΔACP~ΔBCQ.  

 


ΔABC~ΔDEF and their areas are respectively 64 cm2 and 121cm2. If EF = 15.4cm, find BC. 


In Figure 3, ABCD is a trapezium with AB || DC, AB = 18 cm, DC = 32 cm and the distance between AB and DC is 14 cm. If arcs of equal radii 7 cm have been drawn, with centres A,B, C  and D, then find the area of the shaded region.


In  Δ ABC, DE || BC; DC and EB intersects at F. if `"DE"/"BC" = 2/7` , find `("Ar" (triangle "FDE"))/("Ar" (triangle "FBC"))`


In figure , DEF is a right -angled triangle with ∠ E = 90 °.FE is produced to G and GH is drawn perpendicular to DE = 8 cm , DH = 8 cm ,DH = 6 cm and HF = 4 cm , find `("Ar" triangle "DEF")/("Ar" triangle "GHF")`


Δ ABC  ∼ Δ PQR such that AB= 1.5 cm and PQ=2. 1 cm. Find the ratio of areas of Δ ABC and  ΔPQR.


The length of a river in a map is 54cm. if lcm on the map represents 12500m on land, find the length of the river. 


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.


Prove that the area of the triangle BCE described on one side BC of a square ABCD as base is one half of the area of similar triangle ACF described on the diagonal AC as base.


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"`


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.


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"))`


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 1km


Check whether the triangles are similar and find the value of x


If тИЖABC is an isosceles triangle with ∠C = 90° and AC = 5 cm, then AB is


In the figure, if ∠FEG ≡ ∠1 then, prove that DG2 = DE.DF


ΔPQR ~ ΔSUV. Write pairs of congruent angles


ΔABC ~ ΔPQR, A(ΔABC) = 80 sq.cm, A(ΔPQR) = 125 sq.cm, then complete `("A"(Δ"ABC"))/("A"(Δ"PQR")) = 80/125 = (["______"])/(["______"])`, hence `"AB"/"PQ" = (["______"])/(["______"])`


In ΔABC, AP ⊥ BC and BQ ⊥ AC, B−P−C, A−Q−C, then show that ΔCPA ~ ΔCQB. If AP = 7, BQ = 8, BC = 12, then AC = ?


In ΔCPA and ΔCQB

∠CPA ≅ [∠ ______]    ...[each 90°]

∠ACP ≅ [∠ ______]   ...[common angle]

ΔCPA ~ ΔCQB     ......[______ similarity test]

`"AP"/"BQ" = (["______"])/"BC"`    .......[corresponding sides of similar triangles]

`7/8 = (["______"])/12`

AC × [______] = 7 × 12

AC = 10.5


ΔABC and ΔBDE are two equilateral triangles such that D is the mid point of BC. Ratio of the areas of triangle ΔABC and ΔBDE is ______.


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