Let S be a square of unit area. Consider any quadrilateral which has one vertex on each side of S. If `a , b , c and d` denote the lengths of sides of the quadrilateral, prove that `2 ≤ a_2 + b_2 + c_2 + d_2 le 4`

Find the length of the side of a square whose area is 2304 m^2 .

Find the length of the side of a square whose area is 441 m^2 .

The vertices of a quadrilateral are points A (- 4, 3), B (0, 0), C (4, 0) and D (0, 3). Find the lengths of its sides and point out what type of quadrilateral is it ?

A quadrilateral has the vertices at the points (-4, 2), (2, 6) (8, 5) and (9, - 7). Show that the mid-points of the sides of the quadrilateral are the vertices of a parallelogram.

In any quadrilateral ABCD, prove that: sin (A + B) + sin(C +D)=0 .

If a,b,c,d are the side of a quadrilateral, then find the the minimum value of (a^(2)+b^(2)+c^(2))/(d ^(2))

Let PQR be a triangle of area Delta with a = 2, b = 7//2 , and c = 5//2 , where a, b and c are the lengths of the sides of the triangle opposite to the angles at P, Q and R, respectively. Then (2 sin P - sin 2P)/(2 sin P + sin 2P) equals

Let S_1, S_2, be squares such that for each ngeq1, the length of a side of S_n equals the length of a diagonal of S_(n+1)dot If the length of a side of S_1 is 10cm , then the least value of n is the area of S_n less than 1 sq. cm?

A triangle A B C with fixed base B C , the vertex A moves such that cosB+cosC=4sin^2(A/2)dot If a ,b and c , denote the length of the sides of the triangle opposite to the angles A , B ,a n dC , respectively, then (a) b+c=4a (b) b+c=2a (c)the locus of point A is an ellipse (d)the locus of point A is a pair of straight lines

If the angle A ,Ba n dC of a triangle are in an arithmetic propression and if a , ba n dc denote the lengths of the sides opposite to A ,Ba n dC respectively, then the value of the expression a/csin2C+c/asin2A is (a) 1/2 (b) (sqrt(3))/2 (c) 1 (d) sqrt(3)