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Note on Similarity

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Conditions for similarity of triangles

There are three conditions for similarity of triangles:

i) Angle, Angle similarity test

Fig:Angle Angle

Fig:Angle Angle

If two angles of one triangles are respectively equal to two angles of another triangle, then two triangles are similar.

For example:

Here,∠B =∠Y and∠C =∠Z. The remaining angles∠A and∠X are also equal.

∴ \(\triangle\)ABC∼ \(\triangle\)XYZ

 

ii) Side, Side, Side similarity test

Fig: SSS

 

Fig: SSS

If the corresponding sides of two triangles are proportional, then the triangles are similar.

For example:

Here, PQ/XY = QR/YZ = PR/XZ

∴ \(\triangle\)PQR∼ \(\triangle\)XYZ

 

iii) Side, Angle, Side similarity test

Fig: SAS

Fig: SAS

 

 

If two corresponding sides of two triangles are proportional and the angle contained by these sides are equal, then the triangles are similar.

For example:

Here, XY/AB = YZ/BC and∠Y =∠B

∴ \(\triangle\)XYZ∼ \(\triangle\)ABC

 

Similar polygons

Two polygons are similar under following conditions:

i) When two or more polygons are equiangular, they are similar.

In the figure, ∠A =∠P,∠B =∠Q,∠C =∠R,∠D =∠S

∴ quad ABCD∼ quad PQRS

ii) When the corresponding sides of two polygons are proportional, they are similar.

In the figure, AB/PQ = BC/QR = CD/RS = DA/SP

∴ quad ABCD∼ quad PQRS

iii) When the corresponding diagonals of the polygons are proportional to their corresponding sies, they are similar.

 

 

 

In the figure, AC/PR = BD/QS = AB/PQ

∴ quad ABCD∼ quad PQRS

iv) When the corresponding diagonals divide the polygons into the equal number of similar triangles, the polygons are equal.

 

 

 

\(\triangle\)ABC∼ \(\triangle\)PQR, \(\triangle\)ACD∼ \(\triangle\)PRS, \(\triangle\)ADE∼ \(\triangle\)PSV

∴ polygon ABCDE∼ polygon PQRS


Note: Theorem with '*' in similarity chapter do not need proof or experimental verification but the problems related to them are included in the curriculum.

 

 

  • Two geometrical objects are called similar if they both have the same shape, or one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling (enlarging or shrinking), possibly with additional translation, rotation and reflection.
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Very Short Questions

(i) CEA = CBD Given
(ii) ACE = BCD Common Angle
(iii) CAE = BDC Remaining Angle
(iv) AEC~BDC By A.A.A similarity
(v) \(\frac{CE}{BC}\) =\(\frac{CA}{CD}\)
or ,\(\frac{ED + 5}{6}\) =\(\frac{3 + 6}{5}\)
or ,\(\frac{ED + 5}{6}\) =\(\frac{9}{5}\)
or , ED + 5 =\(\frac{54}{5}\)
or , ED = 10.8 -5 = 5.8 cm
Being AEC ~ BDC

Here , DCO and OAB

1. CDO = OBA 1. Being DC || AB , DB is transversal and alternate angles.
2/ DCO = OAB 2. Being DC || AB , AC is transversal and alternate angles.
3/ DCO ~ OAB 3. By A.A.A similarity
4.\(\frac{AB}{DC}\) =\(\frac{BO}{OD}\)
or ,\(\frac{x}{3}\) =\(\frac{5}{2}\)
or , x =\(\frac{15}{2}\) = 7.5
x = 7.5 cm
4. Corresponding sides of similar triangle are proportional

Here , p = 7cm , b = 5cm , h= ?
We know that ,
h2 = p2 + b2

or , h2 = 72 + 52 = 49 + 25 = 74
\(\therefore\) h = \(\sqrt{74}\) = 8.6 cm Ans.

Here , h = 13cm , p = 8cm , b = ?
We know that ,
p2 + b2 = h2
or , 82 + b2 = 132
or , b2 = 132 - 82 = 169 - 64 = 105
b = \(\sqrt{105}\) = 10.25 cm Ans.

Here , GH2 + H I2 = GI2
or , GH2 + 242 = 252
or , GH2 = 252 - 242
or , GH2 = 625 - 57
or , GH2 = 625 - 576
or , GH2 = 49
or , GH= \(\sqrt{49}\) = 7 cm

In given rectangle ABCD , AB = 12cm , BC= 8cm , diagonal (AC) = ?
In rt. angled triangle ABC
AC2 = AB2 + BC2
or , AC2 = 122 + 82 = 144 + 64
= 208
Diagonal AC = \(\sqrt{208}\) = 14.42 cm

In square ABCD , BC = 6cm , AB = 6cm , diagonal(AC) =?
In rt. angled triangle ABC ,
AC2 = Ab2 + BC2
or , AC2 =62 + 62 = 36 + 36 = 72
diagonal AC = \(\sqrt{72}\) =\(\sqrt[6]{2}\) = 8.48 cm.

In rectangle ABCD , AB = 8cm , diagonal AC = 12cm ,BC = ?
In rt. angled triangle ABC
AB2 + BC2 = AC2
or , 82 + BC2 = 122
or , BC2 = 122 - 82 = 144 - 64 = 80
or , BC = \(\sqrt{80}\) = 8.94 cm Ans.

Let us consider AB be the telephone post and CA be wire.
Here CB is the perpendicular distance from the rope fixed at ground C to the post AB.
Here , AB = 7m and AC = 7.6 m
Now , in rt.angled triangle ABC ,
AB2 + BC2= AC2
or , 72 + BC2 = 7.62
or , BC2 = 7.62 - 72 = 57.76 - 49 = 8.76
\(\therefore\) BC = \(\sqrt{8.76}\) = 2.96 m Ans.

In rectangle ABCD ,
Length (AB)= 5.1 cm , diagonal (AC) = 6.1 cm and breadth (BC) = ?
Here , in rt., angled triangle ABC ,
(5.1)2 + (BC)2 = (6.1)2
or , BC2 = (6.1)2 - (5.1)2
or , BC2 = 37.21 - 26.01 = 11.2
\(\therefore\) BC = \(\sqrt{11.2}\) = 3.35 cm
Area of rectangle ABCD = length \(\times\) breadth = 5.1 \(\times\) 3.35 cm 2
= 17.085 cm 2

In DEF , EF =12cm , DE = 9cm , AD = ?
In rt. angled triangle DEF , using pythagoras theorem ,
DF2 = EF2 + DE2
or , DF2 = 122 + 92
or , DF = \(\sqrt{144 + 81}\) = \(\sqrt{225}\) = 15.
Now , in rt. angled triangle ADF ,
AF = 8cm , DF = 15 cm , DA = ?
Here , in ADF ,
DA2 = DF2 + AF2
or , x2 = 152 + 82
or , x2 = 225 + 64
\(\therefore\) x = \(\sqrt{289}\) = 17cm.

Here , AC2 + AB2 = (15)2 + (8)2 = 289
and BC2 = (17)2 = 289
AC2 + BC2 = AB2
h2 = p2 + b2 , so given triangle is right angled triangle in which BAC = 90o

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  • The perimeter of two similar triangles A BC and PQR are 25 cm and 15 cm respectively.If  one side of ( riangle)ABC is 9 cm, find the corresponding side of ( riangle)PQR.

    5.4 cm


    5.5 cm


    2.5 cm


    6.7 cm


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

    5 cm


    10 cm


    20 cm


    15 cm


  • The measures of sides of ( riangle)ABC are 2 cm,4 cm and 5 cm respectively.If ( riangle)ABC ∼( riangle)PQR and the perimeter of ( riangle)PQR = 22 cm,find the measure of sides of ( riangle)PQR.

    2 cm,14 cm,12 cm


    4 cm,8 cm,10 cm


    1 cm,9 cm,13 cm


    5 cm,7 cm,9 cm


  • ( riangle)ABC∼( riangle)PQR and AB=6 cm,BC = 7 cm,CA = 8 cm are given. If ( riangle)PQR has perimeter 42 cm,find its sides.

    12 cm,13.5 cm,18 cm


    11 cm,13.6 cm,11 cm


    9 cm, 5 cm,16 cm


    17 cm,12 cm,20 cm


  • How many conditions are there for similarity of triangles?

    8
    3
    4
    2
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Are a similar triangles congruent to each other?


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