Question

If x= 1+2+3+.......+2000, find the value of x.

2) a wooden article was made by scooping out a hemisphere from each of a solid cylinder. If the height of cylinder is 10cm and base is of radius 3.5cm find the TSA of article.

3) show that for any two positive odd integers a and b , the expression a2+b2 is even but not divisible by 4.

4) if all sides oofa parallalogram touches a circle then prove that the parallalogram is a rhombus.

Answer 1

I know the answer of question 4 only.....

4)
Let ABCD be a parallelogram which circumscribes the circle.
AP = AS [Tangents drawn from an external point to a circle are equal in length]
BP = BQ [Tangents drawn from an external point to a circle are equal in length]
CR = CQ [Tangents drawn from an external point to a circle are equal in length]
DR = DS [Tangents drawn from an external point to a circle are equal in length]
Consider, (AP + BP) + (CR + DR) = (AS + DS) + (BQ + CQ)
AB + CD = AD + BC
But AB = CD and BC = AD  [Opposite sides of parallelogram ABCD]
AB + CD = AD + BC
Hence 2AB = 2BC
Therefore, AB = BC  
Similarly, we get AB = DA and  DA = CD
Thus, ABCD is a rhombus.

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