[tex]\bold{\huge{\underline{\pink{ Solution }}}}[/tex]
Given :-
- Rectangle 1 has length x and width y
- Rectangle 2 is made by multiplying each dimensions of rectangle 1 by a factor of k
- Where, k > 0
Answer 1 :-
Yes, The rectangle 1 and rectangle 2 are similar .
According to the similarity theorem :-
- If the ratio of length and breath of both the triangles are same then the given triangles are similar.
Let's Understand the above theorem :-
The dimensions of rectangle 1 are x and y
Now,
- Rectangle 2 is made by multiplying each dimensions of rectangle 1 by a factor of k .
Let assume the value of K be 5
Therefore,
The dimensions of rectangle 2 are
[tex]\sf{ 5x \:and \:5y }[/tex]
Now, The ratios of dimensions of both the rectangle :-
- [tex]\bold{Rectangle 1 = Rectangle 2}[/tex]
[tex]\bold{\dfrac{ x }{y}}{\bold{ = }}{\bold{\dfrac{5x}{5y}}}[/tex]
[tex]\bold{\blue{\dfrac{ x }{y}}}{\bold{\blue{ = }}}{\bold{\blue{\dfrac{x}{y}}}}[/tex]
From above,
We can conclude that the ratios of both the rectangles are same
Hence , Both the rectangles are similar
Answer 2 :-
Here,
We have to proof that, the
- Perimeter of rectangle 2 = k(perimeter of rectangle 1 )
In the previous questions, we have assume the value of k = 5
Therefore,
According to the question,
Perimeter of rectangle 1
[tex]\sf{ = 2( length + Breath) }[/tex]
[tex]\bold{\pink{= 2( x + y ) }}[/tex]
Thus, The perimeter of rectangle 1
Perimeter of rectangle 2
[tex]\sf{ = 2( length + Breath) }[/tex]
[tex]\sf{ = 2(5x + 5y) }[/tex]
[tex]\sf{ = 2 × 5( x + y) }[/tex]
[tex]\bold{\pink{= 10(x + y) }}[/tex]
Thus, The perimeter of rectangle 2
According to the given condition :-
- Perimeter of rectangle 2 = k( perimeter of rectangle 1 )
Subsitute the required values,
[tex]\sf{ 2(x + y) = 10(x + y)}[/tex]
[tex]\bold{\pink{2x + 2y = 5(2x + 2y) }}[/tex]
From Above,
We can conclude that the, Perimeter of rectangle 2 is 5 times of perimeter of rectangle 1 and we assume the value of k = 5.
Hence, The perimeter of rectangle 2 is k times of rectangle 1
Answer 3 :
Here,
We have to proof that ,
- The area of rectangle 2 is k² times of the area of rectangle 1.
That is,
- Area of rectangle 1 = k²( Area of rectangle)
Therefore,
According to the question,
Area of rectangle 1
[tex]\sf{ = Length × Breath }[/tex]
[tex]\sf{ = x × y }[/tex]
[tex]\bold{\red{= xy }}[/tex]
Area of rectangle 2
[tex]\sf{ = Length × Breath }[/tex]
[tex]\sf{ = 5x × 5y }[/tex]
[tex]\bold{\red{ = 25xy }}[/tex]
According to the given condition :-
- Area of rectangle 1 = k²( Area of rectangle)
[tex]\sf{ xy = 25xy }[/tex]
[tex]\bold{\red{xy = (5)²xy }}[/tex]
From Above,
We can conclude that the, Area of rectangle 2 is (5)² times of area of rectangle 1 and we have assumed the value of k = 5
Hence, The Area of rectangle 2 is k times of rectangle 1 .
