# If x=-3y=-4; x=7y=2 are the four sides of a rectangle. Find the equation of the diagonals.

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The question consist of two parts, each part giving the either directly the dimensions two sides of a rectangle as 'x' and 'y'.

It is then required to find out the equation of the diagonal of the rectangles.

The steps in doing this will be to calculate value of y in terms of x

Then, substitute this value of y in the equation of diagonal. which is:

diagonal = d = (x^2 + y^2)^1/2

Then simplify and convert this equation in to a more convenient form:

**Solution:**

(1) x = -3y = -4

x = -3y

Therefore: y = -x/3

Substituting this value of y in equation for diagonal, that is:

d = (x^2 + y^2)^1/2, we get

d = [(x^2 + -(x/3)^2]^1/2 =

d = [x^2 + (x^2)/9]^1/2

d = [(10/9)x^2)]^1/2 = [(10^1/2)/3]x = 1.0541x

Substituting the given value of x = -4

The value of d we get is

d = 1.0541*4 = 4.2164

(we have not used the -ve sign as it has no impact on the length of the diagonal)

(2) x = 7y = 2

x = 7y

Therefore: y = x/7

Substituting this value of y in equation for diagonal, that is:

d = (x^2 + y^2)^1/2, we get

d = [(x^2 + (x/7)^2]^1/2 =

d = [x^2 + (x^2/49)]^1/2

d = [(50/49)x^2)]^1/2 = [(50^1/2)/7]x = 1.0102x

Substituting the given value of x = 2

The value of d we get is

d = 1.0102*2 = 2.0204

x = -3y = -4 : x=7y=2are the equations of 4 sides. To find the equations of the diagonals.

Solu tion:

The equations of 4 sides are:

x = -4 and y = 4/3 is of pair of adjascent perpendicular sides and

x=2 and y =2/7 are the other pair of perpendicular adjscent sides.

The vertices are say A (-4, 4/3), B(2,4/3) ,C(2,2/7), D(-4, 2/7)

AC and BD are the diagonals.

Equation of the diagonal AC:

y-4/3 = (2/7-4/3)/(2--4)(x---4) Or

y-4/3 = (6-28)/(21(6)) *(x+4) Or

63(y-4/3) = -11(x+4) Or.

11x+63y-84+44 = 0, Or

**11x+63y-40 = 0 is the diagonal AC of the rectangle.**

Diagonal BD:

So the equation of the other diagonal joining B(2,4/3) and D(-4,2/7) is given by :

y-4/3 = (2/7-4/3)/(-4-2) * (x-2) Or

y-4/3 = (11/63)(x-2) Or

63(y-4/3) = 11(x-2) Or

11x-63y+84-22 = 0 Or

**11x-63y+62 = 0 is the equation of the other diagonal BD of the rectangle.**