# The diagonal of rectangle is 12cm. What is the perimeter of the figure formed by joining the mid points of the sides of the rectangle in order?I think the figure formed is Rhombus. Please explain...

The diagonal of rectangle is 12cm. What is the perimeter of the figure formed by joining the mid points of the sides of the rectangle in order?

I think the figure formed is Rhombus.

Please explain with out the use of sin.

sciencesolve | Teacher | (Level 3) Educator Emeritus

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You should come up with the notation for the length and width of rectange such that: 2L expresses the length and 2w expresses the width.

The segment that joins the midpoint of two consecutive sides expresses the hypotenuse of right triangle that has as lengths of legs `(2L)/2`  and `(2w)/2` .

All these right triangles are alike, hence the hypotenuses are equal and the geometric shape that you form linking the midpoints of consecutive segments represents a rhombus.

You may find the length of side of rhombus using pythagorean theorem in a right triangle such that:

`s^2 = L^2 + w^2 =gt s = sqrt(L^2 + w^2)`

The perimeter of rhoumbus is 4 times the side s such that:

`P = 4s =gt P = 4sqrt(L^2 + w^2)`

The problem provides the length of diagonal of rectangle, hence, you may use pythagorean theorem to link this length to the dimension of rectangle.

`4L^2 + 4w^2 = 144`

Dividing by 4 both sides yields:

`L^2 + w^2 = 36 =gt sqrt(L^2 + w^2) = sqrt 36`

`sqrt(L^2 + w^2) = 6`

You should notice that the formula of perimeter of rhombus contains the factor `sqrt(L^2 + w^2), ` hence, substituting 6 for `sqrt(L^2 + w^2)`  yields:

`P = 4*6 cm = 24 cm`

Hence, evaluating the perimeter of rhombus under given conditions yields `P = 24 cm` .

hala718 | High School Teacher | (Level 1) Educator Emeritus

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Let the sides of the rectangle be x and y.

==> x^2 + y^2 = 12^2

==> x^2 + y^2 = 144

Now, we need to find the perimeter of the shape formed  by joining midpoints of the rectangle.

Each side of the new shape ( Rhombus) is the hypotenuse of a right triangle whose sides are (x/2) and (y/2)

Let the sides be h.

==> h^2 = (x/2)^2 + (y/2)^2 = (x^2+y^2)/4

But x^2 + y^2 = 144

==> h^2 = 144/4 = 36

==> h= 6

Then, each side of the rhombus is 6 cm.

==> Then the perimeter is P= 4*6 = 24 cm