Explain why the triangles are similar. then find the distance represented by x

Q- 8 ) The value of x  is 48 feet

Q-9 ) The value of x  is 10  

Step-by-step explanation:

Given two figures as :

8 ) Let AB = 3 ft

          AO = 5 ft

         PQ = x ft

         PO = 80 ft

Now, From figure ,

  AB║PQ

∠ AOB = ∠ POQ  ( vertically opposite angles )

∠ A = ∠ P

So , Δ AOB  Δ POQ

Now From similar triangle property

  =  

Or,   =  

Or, x =  

∴   x = 48 feet

9 ) Now, From the figure

 ABC is a large Triangle and  ADE is a small triangle

So, To prove both are similar

Since  DE ║ BC

∠ ADE = ∠ ABC                     corresponding angles

and , ∠ AED = ∠ ACB            corresponding angles

So, by angle angle

Δ ADE  Δ ABC

Now, From the similarity property

  =  

Or,   =  

Or,  6 × ( 2 x + 4 ) = ( x - 1 ) × (6 + x )

Or,  12 x + 24 = 6 x + x² - 6 - x

Or,  12 x - 5 x + 30 = x²

or,  x² - 7 x - 30 = 0

Or, x² - 10 x + 3 x - 30 = 0

or, x ( x - 10 ) + 3 ( x - 10 ) = 0

Or , ( x - 10 ) ( x + 3 ) = 0

∴  x = 10 , - 3

so, positive value of x is consider , i.e x = 10

Hence The value of x for Q- 8 is 48 feet

And the value of x for Q-9 is 10   .  Answer

answer: no, but if it does it's a obvious pattern.

step-by-step explanation: