1) In a rhombus ABCD, AB=18 and AC=28. Find the area of the rhombus to the nearest tenth.

2) The area of a trapezoid is 190 in.squared. if the height is 11 in. and the length base is 22 in. what is the length of the shorter base. Round to the nearest tenth.

3) Chose the statement that is NOT ALWAYS true. For any parallelogram _________. Is it A) the diagonals are perpendicular B) opposite sides are congruent or C) opposite angles are congruent 

4)(2,3) and (3,1) are opposite vertices in a parallelogram. If (0,0) is the third vertex then the forth vertex is_______.

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1 Answer

  1. The sides of a rhombus are all equal and the diagonals cross at right angles and bisect each other. If the diagonals are drawn the rhombus can be seen to consist of 4 right-angled triangles, with the sides of the rhombus being the hypotenuses. The side of the rhombus is 18 and the length of one diagonal is 28 so the four triangles have a height of 28/2=14 and a hypotenuse of length 18. The third side=sqrt(18^2-14^2)=sqrt(128)=8sqrt(2)=11.3137. The area of each triangle is 0.5*8sqrt(2)*14=56sqrt(2), so the area of the rhombus is 4*56sqrt(2)=224sqrt(2)=316.8.
  2. The area of the trapezoid is made up of a central rectangle and two triangles, one on each side of the rectangle. If the length of the shorter side is x then the area of the rectangle is 11x. Let the base of one of the triangles be b then the base of the other will be 22-x-b. The combined area of the triangles is (11b+11(22-x-b))/2=11(22-x)/2. The combined area of the triangles and rectangle is 11x+11(22-x)/2=190. So 22x+231-11x=380, 11x=380-231=149 and x=149/11=13.5 approx.
  3. A. The diagonals of a parallelogram are not always perpendicular.
  4. Call the parallelogram ABCD where A is (2,3), C is (3,1) and D is (0,0). To get from D to C we go along 3 and up 1. To go from A to B we do the same, so B is (5,4).

 

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