FINDING MISSING SIDE IN SIMILAR TRIANGLES

For the following figures, establish that a pair of triangles is similar, hence find x :

Example 1 :

Solution :

Given, DE || BC

Then, DE/BC  =  AE/AC

AE  =  1 cm, AC  =  5 cm, DE  =  x cm, BC  =  6 cm

x/6  =  1/5

x  =  6/5

x  =  1.2

So, the value of x is 1.2 cm

Example 2  :

Solution :

Given, DE || BC

Then, AD/DB  =  AE/EC

AD  =  x cm, DB  =  9 cm, AE  =  12 cm, EC  =  10 cm

x/9  =  12/10

x/9  =  6/5

5x  =  54

x  =  54/5

x  =  10.8

So, the value of x is 10.8 cm

Example 3  :

Solution :

By considering the small and larger triangles.

Then, DE/AB  =  EC/BC

AB  =  6 cm, DE  =  x cm, EC  =  5 cm, BC  =  9 cm

x/6  =  5/9

x  =  30/9

x  =  10/3

x  =  3 1/3

So, the value of x is 3 1/3 cm

Example 4  :

Solution :

EA/CA  =  ED/BC  =  DA/AB

EA/CA  =  DA/AB

x/(7+x)  =  6/(6+4)

x/(7+x)  =  6/10

10x  =  6(7+x)

10x  =  42+6x

4x  =  42

x  =  42/4

x  =  10.5

So, the value of x is 10.5 cm

Example 5  :

Solution :

In Δ ABC, and ΔCED

<ACB  =  <ECD  (vertically opposite angles are equal)

<ABC  =  <CED (90 degree)

Using AA theorem, the above triangles Δ ABC ~ ΔCED

AB/ED  =  AC/CD  =  BC/DE

2/3  =  8/x

x  =  8(3)/2

x  =  12 cm

Example 6 :

Solution :

In Δ ABC, and ΔCED

Then, DE/AB  =  EC/BC

AB  =  7 cm, BC  =  5 cm, DE  =  4 cm, EC  =  x cm

4/7  =  x/5

x  =  20/7

x  =  2 6/7

So, the value of x is 2 6/7 cm

Example 7 :

Solution :

In Δ ABC, and ΔAED

Given, DE || BC

If <ADE  =  <ABC

<EAD  =  <CAB  

So, ∆ADE ~ ∆ABC

Then, DE/BC  =  AD/AB

AD  =  2 cm, AB  =  5 cm, DE  =  3 cm, BC  =  x cm

3/x  =  2/5

2x  =  15

x  =  15/2

x  =  7.5

So, the value of x is 7.5 cm

Example 8 :

Solution :

In Δ ADE, and ΔABC

<ADE  =  <ABC  (A)

<DAE  =  <BAC  (A)

So, ∆ADE ~ ∆ABC

AE/AC  =  DE/BC  =  DA/AB

x/(x+5)  =  3/6

x/(x+5)  =  1/2

2x  =  x+5

x  =  5

So, the value of x is 5 cm

Example 9  :

Solution :

In Δ CDE, and ΔABC

<CED  =  <CBA

<DCE  =  <ACB

So, ∆DEC ~ ∆ABC

Then, DE/AB  =  EC/BC

AB  =  5 cm, DE  =  2 cm, EC  =  x cm, BC  =  2x+3 cm

2/5  =  x/(2x+3)

5x  =  2(2x+3)

5x  =  4x+6

5x – 4x  =  6

x  =  6

So, the value of x is 6 cm.

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