Question

2.The diagonals of a rhombus are in the ratio 5:12. If its perimeter is 104 CM, findthe lengths of the sides and the diagonals.​

Answer 1

Lenghts of the sides:
`26\ cm`

Lenghts of the diagonals:
`48\ cm` and
`20\ cm`

Explanation:

Look at the rhombus ABCD shown attached, where AC and BD de diagonals of the rhombus.

The sides of a rhombus have equal lenght. Then, since the perimeter of this one is 104 centimeters, you can find the lenght of each side as following:

`AB=BC=CD=DA=(104\ cm)/(4)= 26\ cm`

You know that the diagonals are in the ratio
`5:12`

Then, let the diagonal AC be:

`AC=12x`

This means that AE is:

`AE=(12x)/(2)=6x`

And let the diagonal BD be:

`BD=5x`

So BE is:

`BE=(5x)/(2)=2.5x`

Since the diagonals of a rhombus are perpendicular to each other, four right triangles are formed, so you can use the Pythagorean Theorem:

`a^2=b^2+c^2`

Where "a" is the hypotenuse and "b" and "c" are the legs.

In this case, you can choose the triangle ABE. Then:

`a=AB=26\\b=AE=6x\\c=BE=2.5x`

Substituting values and solving for "x", you get:

`26^2=(6x)^2+(2.5x)^2\\\\676=36x^2+6.25x^2\\\\\sqrt{(676)/(42.25)}=x\\\\x=4`

Therefore, the lenghts of the diagonals are:

`AC=12(4)\ cm=48\ cm`

`BD=5(4)\ cm=20\ cm`