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.
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
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
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