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1.A rectangle and a square have the same perimeter. One side-length of the rectangle is  longer than the other. What is the ratio between the areas of the rectangle and the square?

 

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

 

3.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.

asdfasdfasdf  Feb 7, 2018
edited by asdfasdfasdf  Feb 7, 2018
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2.In rhombus ABCD, points E, F, G and H are the midpoints of AB, BC, CD, DA and  respectively. Quadrilateral EFGH has area 14 and perimeter 16. Find the side length for rhombus ABCD.

 

The side length will = the diagonal length of the quadrilateral

 

So

 

Area  of quadrilateral LW  = 14    (1)

 

Perimeter  of quadrilateral =  16  = 2(L + W)  ⇒ 8 = L + W ⇒  L = 8 - W    (2)

 

Sub (2)  into (1)

 

(8 - W) W =  14   simplify  and rearrange

 

W^2  - 8W  + 14  = 0

 

Solving this for W  gives us both dimensions  of

 

4 + √2  units    and  4  - √2  units

 

And the length of a side of the rhombus is

 

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

 

√  [  16 + 2 + 16 + 2 ]  = √36    =   6 units

 

 

cool cool cool

CPhill  Feb 7, 2018

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