Question

Help!! Explain why the triangles are similar. Then find the distance represented by x

Answer 1

Q- 8 ) The value of x is 48 feet

Q-9 ) The value of x is 10

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
`\sim` Δ POQ

Now From similar triangle property

`(AB)/(PQ)` =
`(AO)/(PO)`

Or,
`(30)/(x)` =
`(50)/(80)`

Or, x =
`(2400)/(50)`

∴ 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
`\sim` Δ ABC

Now, From the similarity property

`(AE)/(AC)` =
`(DE)/(BC)`

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

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