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Savatey [412]
2 years ago
7

Circle O and triangle OQR are shown below. What is the length of OQ?

Mathematics
1 answer:
zysi [14]2 years ago
3 0

Answer:

Step-by-step explanation:

A circle is inscribed in an equilateral triangle PQR with centre O. If angle OQR = 30°, what is the perimeter of the triangle?

This is a circle inscribed in an equilateral triangle with side s.

If you are asking for the perimeter of PQR, it is 3s.

If you are asking for the perimeter of OQR, it is (3+23–√3)s

Since OR and SR are the hypotenuses of right triangles with adjacent side equal to ½ s, their length is ½s / cos 30° = (√3) /3.

(3/3)s + ((√3) /3)s + ((√3) /3)s = ((3 + 2√3)/3)s ≈ 2.1547s

Hope it helps

help me by marking as brainliest....

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Mars2501 [29]
Answers:  
_____________________________________________________
   Part A)  " (3x + 4) " units  . 
_____________________________________________________
   Part B)  "The dimensions of the rectangle are:

                             " (4x + 5y) " units ;  <u>AND</u>:  " (4x − 5y)"  units."
_____________________________________________________

Explanation for  Part A):
_____________________________________________________

Since each side length of a square is the same; 
   
    Area = Length * width = L * w ;  L = w  = s = s ;

      in which:  " s = side length" ;

So, the Area of a square, "A"  = L * w = s * s = s² ;

{<u>Note</u>:  A "square" is a rectangle with 4 (four) equal sides.}.

→  Each side length, "s", of a square is equal.

Given:  s² = "(9x² + 24x + 16)" square units ;

Find "s" by factoring: "(9x² + 24x + 16)" completely:

   →  " 9x² + 24x + 16 ";

Factor by "breaking into groups" :

"(9x² + 24x + 16)"  = 

    →  "(9x² + 12x) (12x + 16)" ;
_______________________________________________________

Given:   " (9x² + 24x + 16) " ; 
_______________________________________________________
Let us start with the term:
_______________________________________________________

" (9x² + 12x) " ; 

    →  Factor out a "3x" ;  → as follows:
_______________________________________

    → " 3x (3x + 4) " ; 

Then, take the term:
_______________________________________
    → " (12x + 16) " ;

And factor out a "4" ;   →  as follows:
_______________________________________

    → " 4 (3x + 4) " 
_______________________________________
We have:

" 9x² + 24x + 16 " ;

    =  " 3x (3x + 4)  +  4(3x + 4) " ;
_______________________________________
Now, notice the term:  "(3x + 4)" ; 

We can "factor out" this term:

3x (3x + 4)  +  4(3x + 4)  = 

     →  " (3x + 4) (3x + 4) " .  → which is the fully factored form of:

                                                   " 9x² + 24x + 16 "  ; 
____________________________________________________
     →  Or; write:  "  (3x + 4) (3x + 4)" ; as:  " (3x + 4)² " .
____________________________________________________
     →  So,  "s² = 9x² + 24x + 16 " ; 

Rewrite as:  " s² = (3x + 4)² " .

     →  Solve for the "positive value of "s" ; 

     →  {since the "side length of a square" cannot be a "negative" value.}.
____________________________________________________
     →  Take the "positive square root of EACH SIDE of the equation; 
              to isolate "s" on one side of the equation; & to solve for "s" ;

     →  ⁺√(s²)  =  ⁺√[(3x + 4)²]   '

To get:

     →  s  = " (3x + 4)" units .
_______________________________________________________

Part A):  The answer is:  "(3x + 4)" units.
____________________________________________________

Explanation for Part B):

_________________________________________________________<span>

The area, "A" of a rectangle is:

    A = L * w ;  

 in which "A" is the "area" of the rectangle;
                "L" is the "length" of the rectangle; 
                "w" is the "width" of the rectangle; 
_______________________________________________________
  Given:  " A = </span>(16x² − 25y²) square units" ;  
   
       →  We are asked to find the dimensions, "L" & "w" ;
       →  by factoring the given "area" expression completely:
____________________________________________________
  → Factor:  " (16x² − 25y²) square units " completely '

Note that:  "16" and: "25" are both "perfect squares" ;
      
We can rewrite: " (16x² − 25y²) "  ;   as:

       =   " (4²x²)  −  (5²y²) " ; and further rewrite the expression:
________________________________________________________
Note:  
________________________________________________________
" (16x²) " ;  can be written as:  "(4x)² " ;

 ↔ " (4x)²  =  "(4²)(x²)" = 16x² "


Note:  The following property of exponents:

         →  (xy)ⁿ = xⁿ yⁿ ;    →  As such:  " 16x² = (4²x²) = (4x)² " . 
_______________________________________________________
Note:
_______________________________________________________

     →   " (25x²) " ;  can be written as:  " (5x)² " ; 

        ↔   "( 5x)²  =  "(5²)(x²)" = 25x² " ; 

Note:  The following property of exponents:

         →  (xy)ⁿ = xⁿ yⁿ ;    →  As such:  " 25x² = (5²x²) = (5x)² " . 
______________________________________________________

→  So, we can rewrite:  " (16x² − 25y²) " ;  

as:  " (4x)² − (5y)² " ;   
 
    → {Note:  We substitute: "(4x)² "  for "(16x²)" ; & "(5y)² "  for "(25y²)" .} . ; 
_______________________________________________________
→  We have:  " (4x)² − (5y)² " ;

→  Note that we are asked to "factor completely" ; 

→  Note that:  " x² − y² = (x + y) (x − y) " ;

      → {This property is known as the "<u>difference of squares</u>".}.

→ As such:  " (4x)² − (5y)² " = " (4x + 5y) (4x − 5y) " .
_______________________________________________________
Part B):  The answer is:  "The dimensions of the rectangle are:

                              " (4x + 5y) " units ;  AND:  " (4x − 5y)"  units."
_______________________________________________________
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