+18 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.
+128460 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
