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1)Square ABCD and rectangle ACEG have the same area. Find the ratio AC:CE.

 

2)A rectangle and a square have the same perimeter. One side-length of the rectangle is 25% longer than the other. What is the ratio between the areas of the rectangle and the square?
Write your answer as a common fraction (or an integer).

 

3) In rhombus ABCD, points E,F,G and H are the midpoints of AB, BC, CD and DA respectively. Quadrilateral EFGH has area 14 and perimeter 16. Find the side length for rhombus ABCD.

 

4)Which of the following statements are correct?

A. If one interior angle of a parallelogram is a right angle, then the parallelogram must be a rectangle.

B. If two diagonals of a rectangle are perpendicular, then the rectangle must be a square.

C. If two diagonals of a rhombus are equal, then the rhombus must be a square.

D. If one interior angle of a rhombus is a right angle, then the rhombus must be a square.

E. If two diagonals of a parallelogram are equal, then the parallelogram must be a rectangle.

Enter all correct options as a list of letters, separated by commas.

Guest Jan 8, 2018
 #1
avatar+88871 
+1

1)Square ABCD and rectangle ACEG have the same area. Find the ratio AC : CG

 

Let the side of the square  = s

 

Then  AC = √2s

 

The area of the square  is  s2

 

The  area  of the rectangle  is  also  s2  = AC * CE

 

So

 

s2  =  √2s *  CE    ⇒  CE  =   s / √2

 

So.........

 

 

CG   =   √  [ AC2  + CE2 ]    =   √  [ 2s2  + s2 / 2  ]   =  √  [ 5s2 / 2 ]  =  s √ [ 5/2]

 

So    AC  : CE   =  

 

AC  / CE   =      √2s  /  [ s  √ [ 5/2 ]  ]    =  2 / √5

 

 

cool cool cool

CPhill  Jan 9, 2018
 #2
avatar+88871 
+1

2)A rectangle and a square have the same perimeter. One side-length of the rectangle is 25% longer than the other. What is the ratio between the areas of the rectangle and the square?
Write your answer as a common fraction (or an integer).

 

 

Perimeter of the square is  4s     where s  is the side length

 

And the area of the square  is   s2

 

Half the perimeter of the rectangle   = 2s

 

So....  W  +  L   =   2s

Let  W  = 1.25L

So.....  [ W  + L ]  = 2s    ⇒  [ 1.25L  + L ]   =  2s  ⇒  2.25L = 2s  ⇒

L =  (2/2.25)s ⇒  L  =  (4/5)s

So  W   =  1.25L  =   (5/4)L  =  (5/4)(4/5)s  =  s

 

And the area of the rectangle  is  L * W  =   (4/5)s * s  =  (4/5)s2

 

So...

 

area of rectangle                    (4/5)s2                      4

______________      =           ______        =          __     

area of square                            s2                         5

 

 

cool cool cool

CPhill  Jan 9, 2018
 #3
avatar+88871 
+1

3) In rhombus ABCD, points E,F,G and H are the midpoints of AB, BC, CD and DA respectively. Quadrilateral EFGH has area 14 and perimeter 16. Find the side length for rhombus ABCD.

 

Let  the sides of the quadrilateral  be  W  and  L

And  W * L  =  14  ⇒   L  = 14 / W

 

And

 

2 ( W + L )  =  16

 

W + L   =  8    ⇒  W  +  14/ W   =  8   ⇒  W^2  + 14  =  8W   ⇒

 

W^2   -  8W  +  14   = 0

 

Solving this for W  we have that  W  =  4 + √2     and  L  =  4 - √2

 

The  side length  of  the rhombus  =   the length of a diagonal of the quadrilateral  = 

 

√  [   ( 4 + √2)2  +  (4  - √2)2  ]   =

 

√  [ 16 + 2  +  16 + 2  ]  =

 

√36   =

 

6 units

 

 

cool cool cool

CPhill  Jan 9, 2018

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