Let ∆ Abc ∽ ∆ Def and Their Areas Be Respectively, 64 Cm2 and 121 Cm2. If Ef = 15⋅4 Cm, Find Bc. - Mathematics

Advertisements
Advertisements
Sum

Let ∆ ABC ∽ ∆ DEF and their areas be respectively, 64 cm2 and 121 cm2. If EF = 15⋅4 cm, find BC.

Advertisements

Solution

Given: ABC ~ DEF
We know the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
`("arΔABC")/("arΔDEF") = ("BC"/"EF")^2`

⇒ `64/121 = ("BC"/15.4)^2`

⇒ `(8/11)^2 = ("BC"/15.4)^2`

⇒ `8/11 = "BC"/15.4`

⇒ BC = `(8 xx 15.4)/11` = 11.2 cm
Thus, BC = 11.2 cm.

  Is there an error in this question or solution?
2018-2019 (March) 30/4/3

RELATED QUESTIONS

In the following figure, in Δ PQR, seg RS is the bisector of ∠PRQ.

PS = 3, SQ = 9, PR = 18. Find QR.


In the figure given below, Ray PT is bisector of ∠QPR. If PQ = 5.6 cm, QT = 4 cm and TR = 5 cm, find the value of x .


In a triangle ABC, line l || Side BC and line l intersects side AB and AC in points P and Q, respectively. Prove that: `"AP"/"BP"="AQ"/"QC"`


The perimeters of two similar triangles ABC and PQR are respectively 36 cm and 24 cm. If PQ = 10 cm, find AB


In figure, `\frac{AO}{OC}=\frac{BO}{OD}=\frac{1}{2}` and AB = 5 cm. Find the value of DC.


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 figure, ∠A = ∠CED, prove that ∆CAB ~ ∆CED. Also, find the value of x.


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

PE = 3.9 cm, EQ = 3 cm, PF = 3.6 cm and FR = 2.4 cm


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


Using Basic proportionality theorem, prove that a line drawn through the mid-points of one side of a triangle parallel to another side bisects the third side. (Recall that you have proved it in Class IX).


The diagonals of a quadrilateral ABCD intersect each other at the point O such that `(AO)/(BO) = (CO)/(DO)`  Show that ABCD is a trapezium


Show that ΔABC, where A(–2, 0), B(2, 0), C(0, 2) and ΔPQR where P(–4, 0), Q(4, 0), R(0, 2) are similar triangles


In the following figure, seg DH ⊥ seg EF and seg GK ⊥ seg EF. If DH = 18 cm, GK = 30 cm and `A(triangle DEF) = 450 cm^2`, then find:

1) EF

2) `A(triangle GFE)`

3) `A(square DFGE)`


Given `triangle ABC ~ triangle PQR`, if `(AB)/(PQ) = 1/3`, then find `(ar  triangle ABC)/(ar triangle PQR)`


In ΔABC and ΔDEF, it is being given that: AB = 5 cm, BC = 4 cm and CA = 4.2 cm; DE=10cm, EF = 8 cm and FD = 8.4 cm. If AL ⊥ BC and DM ⊥ EF, find AL: DM.


True or False:

all equiangular triangles are similar.


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


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


(i) Name the three pairs of similar triangles.
(ii) Find the lengths of EC and EF.


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

(i) Determine the ratios `("AD")/("AB") and ("DE")/("BC")` 

(ii) Prove that ∆DEF is similar to ∆CBF Hence, find `("EF")/("FB").`

(iii) What is the ratio of the areas of ∆DEF and ∆BFC.


A model of a ship if made to a scale of 1 : 200.
(i) Thelength of the model is 4 m; calculate the length of the ship. 

(ii) The area of the deck of the ship is 160000 m2; find the area of the deck of the model.
(iii) The volume of the model is 200 litres; calculate the volume of the ship in m3.


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: 


In the given figure, ΔOAB ~ ΔOCD. If AB = 8cm, BO = 6.4cm, OC = 3.5cm and CD = 5cm, find (i) OA (ii) DO.  


The areas of two similar triangles ABC and PQR are in the ratio 9:16. If BC = 4.5cm, find the length of QR. 


ΔABC~ΔPQR and ar(ΔABC) = 4, ar(ΔPQR) . If BC = 12cm, find QR. 


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


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


In the given figure, X is any point in the interior of triangle. Point X is joined to vertices of triangle. Seg PQ || seg DE, seg QR || seg EF. Fill in the blanks to prove that, seg PR || seg DF. 

Proof :  In ΔXDE, PQ || DE         ...`square`

∴ `"XP"/square = square/"QE"`                               ...(I) (Basic proportionality theorem)

In ΔXEF, QR || EF                       ...`square`

∴ `square/square = square/square                                                           ..."(II)" square`

∴ `square/square = square/square`                                ...from (I) and (II)

∴ seg PR || seg DF           ...(converse of basic proportionality theorem)


In the given figure, ∠ABC = 75°, ∠EDC = 75° state which two triangles are similar and by which test? Also write the similarity of these two triangles by a proper one to one correspondence.


Select the appropriate alternative.
In ∆ABC and ∆PQR, in a one to one correspondence \[\frac{AB}{QR} = \frac{BC}{PR} = \frac{CA}{PQ}\] 


 In ∆ABC and ∆DEF ∠B = ∠E, ∠F = ∠C and AB = 3DE then which of the statements regarding the two triangles is true ?


In the given figure, seg XY || seg BC, then which of the following statements is true?


In the given figure, A – D– C and B – E – C seg DE || side AB If AD = 5, DC = 3, BC = 6.4 then Find BE.

 


The dimensions of a buiIding are 50 m Iong, 40m wide and 70m high. A model of the same building is made with a scale factor of 1: 500. Find the dimensions of the model.


A ship is 400m laig and 100m wide. The length of its model is 20 cm. find the surface area of the deck of the model. 


If ΔABC ~ ΔPQR  and ∠A = 60°, then ∠P = ?


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. 


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.

 


Construct a ΔABC in which CA = 6 cm, AB = 5 cm and ∠BAC = 45°. Then construct a triangle whose sides are `3/5` of the corresponding sides of ΔABC.


Points A(3, 1), B(5, 1), C(a, b) and D(4, 3) are vertices of a parallelogram ABCD. Find the values of a and b.


Prove that, if a line is drawn parallel to one side of a triangle to intersect the other two sides, then the two sides are divided in the same ratio.


In ΔPQR, L and M are two points on the base QR, such that ∠LPQ = ∠QRP and ∠RPM = ∠RQP.
Prove that : (i) ΔPQL ∼ ΔRPM
(ii) QL. Rm = PL. PM
(iii) PQ2 = QR. QL.


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 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"`.


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?


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


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 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


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.


A girl looks the reflection of the top of the lamp post on the mirror which is 6.6 m away from the foot of the lamppost. The girl whose height is 1.25 m is standing 2.5 m away from the mirror. Assuming the mirror is placed on the ground facing the sky and the girl, mirror and the lamppost are in the same line, find the height of the lamp post.


In the adjacent figure, ∆ABC is right angled at C and DE ⊥ AB. Prove that ∆ABC ~ ∆ADE and hence find the lengths of AE and DE


If figure OPRQ is a square and ∠MLN = 90°. Prove that QR2 = MQ × RN


If ∆ABC ~ ∆DEF such that area of ∆ABC is 9 cm2 and the area of ∆DEF is 16 cm2 and BC = 2.1 cm. Find the length of EF.


In ∆LMN, ∠L = 60°, ∠M = 50°. If ∆LMN ~ ∆PQR then the value of ∠R is


The perimeters of two similar triangles ∆ABC and ∆PQR are 36 cm and 24 cm respectively. If PQ = 10 cm, then the length of AB is 


In the adjacent figure ∠BAC = 90° and AD ⊥ BC then
 


If BD ⊥ AC and CE ⊥ AB, prove that `"CA"/"AB" = "CE"/"DB"`


D is the mid point of side BC and AE ⊥ BC. If BC = a, AC = b, AB = c, ED = x, AD = p and AE = h, prove that c2 = `"p"^2 - "a"x + "a"^2/4`


D is the mid point of side BC and AE ⊥ BC. If BC = a, AC = b, AB = c, ED = x, AD = p and AE = h, prove that b2 + c2 = `2"p"^2 + "a"^2/2`


In any triangle _______ sides are opposite to equal angles


From the given figure, prove that ΔABC ~ ΔEDF


In the given figure YH || TE. Prove that ΔWHY ~ ΔWET and also find HE and TE


In the given figure, if ΔEAT ~ ΔBUN, find the measure of all angles.


In the given figure, UB || AT and CU ≡ CB Prove that ΔCUB ~ ΔCAT and hence ΔCAT is isosceles.


If in triangles PQR and XYZ, `"PQ"/"XY" = "QR"/"ZX"` then they will be similar if


A flag pole 15 m high casts a shadow of 3 m at 10 a.m. The shadow cast by a building at the same time is 18.6 m. The height of the building is


If ∆ABC – ∆PQR in which ∠A = 53° and ∠Q = 77°, then ∠R is


From the figure, prove that ∆SUN ~ ∆RAY


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


Given ΔABC ~ ΔDEF, if ∠A = 45° and ∠E = 35° then ∠B = ?


Are triangles in figure similar? If yes, then write the test of similarity.


If ΔABC ~ ΔLMN and ∠B = 40°, then ∠M = ? Give reason.


In fig. BP ⊥ AC, CQ ⊥ AB, A−P−C, and A−Q−B then show that ΔAPB and ΔAQC are similar.

In ΔAPB and ΔAQC

∠APB = [   ]°     ......(i)

∠AQC = [   ]°  ......(ii)

∠APB ≅ ∠AQC    .....[From (i) and (ii)]

∠PAB ≅ ∠QAC    .....[______]

ΔAPB ~ ΔAQC     .....[______]


Observe the figure and complete the following activity


In fig, ∠B = 75°, ∠D = 75°

∠B ≅ [ ______ ]       ...[each of 75°]

∠C ≅ ∠C         ...[ ______ ]

ΔABC ~ Δ [ ______ ]     ...[ ______ similarity test]


Areas of two similar triangles are 225 cm2 and 81 cm2. If side of smaller triangle is 12 cm, find corresponding side of major triangle


There are two poles having heights 8 m and 4 m on plane ground as shown in fig. Because of sunlight shadows of smaller pole is 6m long, then find the length of shadow of longer pole.


In given fig., quadrilateral PQRS, side PQ || side SR, AR = 5 AP, then prove that, SR = 5PQ


In the figure PQ || BC. If `"PQ"/"BC" = 2/5` then `"AP"/"PB"` is ______.


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 ______.


In a square of side 10 cm, its diagonal = ______.


In the given figure, ΔABC ∼ ΔQPR, If AC = 6 cm, BC = 5 cm, QR = 3 cm and PR = x; them the value of x is ______.


In ΔABC, PQ || BC. If PB = 6 cm, AP = 4 cm, AQ = 8 cm, find the length of AC.


In figure, if AD = 6 cm, DB = 9 cm, AE = 8 cm and EC = 12 cm and ∠ADE = 48°. Find ∠ABC. 


In ΔPQR, S and T are points on PQ and PR respectively. `(PS)/(SQ) = (PT)/(TR)` and ∠PST = ∠PRQ. Prove that PQR is an isosceles triangle.


In ΔABC, DE || BC (as shown in the figure), If AD = 4 cm, AB = 9 cm and AC = 13.5 cm, then the length of EC is ______.


Share
Notifications



      Forgot password?
Use app×