The vertices of quadrilateral abcd are a(1, a - 3), b(10, a), c(9, a + 3) and d(0, a).prove that the quadrilateral is a rectangle. find the midpoints of the diagonals. the midpoint of ac is (5,1)the midpoint of bd is (5,1)find the lengths of the diagonals.

Answer with explanation:

The Vertices of quadrilateral A B CD, are A(1, a - 3), B(10, a), C(9, a + 3) and D(0, a).

Mid point of AC= (5,1)

Mid point of B D= (5,1)

Mid point formula of two points having coordinates , (a,b) and (c,d) is , if (x,y), then

   

 

So, the coordinates of vertices of quadrilateral A B CD, are A(1, -2), B(10, 1), C(9, 4) and D(0, 1).

Distance formula of two points having coordinates, (a,b) and (c,d) is,

       

   

Opposite sides  AB and CD are equal to 3√10 unit and , BC and AD are equal to √10 unit.

Also,length of Diagonals , AC=B D=10 unit.

∴ The Quadrilateral, AB CD is a Rectangle.

P(chicago, illinois)

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