Answer:
Lenghts of the sides: ![26\ cm](https://tex.z-dn.net/?f=26%5C%20cm)
Lenghts of the diagonals:
and ![20\ cm](https://tex.z-dn.net/?f=20%5C%20cm)
Step-by-step 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=\frac{104\ cm}{4}= 26\ cm](https://tex.z-dn.net/?f=AB%3DBC%3DCD%3DDA%3D%5Cfrac%7B104%5C%20cm%7D%7B4%7D%3D%2026%5C%20cm)
You know that the diagonals are in the ratio ![5:12](https://tex.z-dn.net/?f=5%3A12)
Then, let the diagonal AC be:
![AC=12x](https://tex.z-dn.net/?f=AC%3D12x)
This means that AE is:
![AE=\frac{12x}{2}=6x](https://tex.z-dn.net/?f=AE%3D%5Cfrac%7B12x%7D%7B2%7D%3D6x)
And let the diagonal BD be:
![BD=5x](https://tex.z-dn.net/?f=BD%3D5x)
So BE is:
![BE=\frac{5x}{2}=2.5x](https://tex.z-dn.net/?f=BE%3D%5Cfrac%7B5x%7D%7B2%7D%3D2.5x)
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](https://tex.z-dn.net/?f=a%5E2%3Db%5E2%2Bc%5E2)
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](https://tex.z-dn.net/?f=a%3DAB%3D26%5C%5Cb%3DAE%3D6x%5C%5Cc%3DBE%3D2.5x)
Substituting values and solving for "x", you get:
![26^2=(6x)^2+(2.5x)^2\\\\676=36x^2+6.25x^2\\\\\sqrt{\frac{676}{42.25}}=x\\\\x=4](https://tex.z-dn.net/?f=26%5E2%3D%286x%29%5E2%2B%282.5x%29%5E2%5C%5C%5C%5C676%3D36x%5E2%2B6.25x%5E2%5C%5C%5C%5C%5Csqrt%7B%5Cfrac%7B676%7D%7B42.25%7D%7D%3Dx%5C%5C%5C%5Cx%3D4)
Therefore, the lenghts of the diagonals are:
![AC=12(4)\ cm=48\ cm](https://tex.z-dn.net/?f=AC%3D12%284%29%5C%20cm%3D48%5C%20cm)
![BD=5(4)\ cm=20\ cm](https://tex.z-dn.net/?f=BD%3D5%284%29%5C%20cm%3D20%5C%20cm)