Answer:
Lenghts of the sides: [tex]26\ cm[/tex]
Lenghts of the diagonals: [tex]48\ cm[/tex] and [tex]20\ cm[/tex]
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:
[tex]AB=BC=CD=DA=\frac{104\ cm}{4}= 26\ cm[/tex]
You know that the diagonals are in the ratio [tex]5:12[/tex]
Then, let the diagonal AC be:
[tex]AC=12x[/tex]
This means that AE is:
[tex]AE=\frac{12x}{2}=6x[/tex]
And let the diagonal BD be:
[tex]BD=5x[/tex]
So BE is:
[tex]BE=\frac{5x}{2}=2.5x[/tex]
Since the diagonals of a rhombus are perpendicular to each other, four right triangles are formed, so you can use the Pythagorean Theorem:
[tex]a^2=b^2+c^2[/tex]
Where "a" is the hypotenuse and "b" and "c" are the legs.
In this case, you can choose the triangle ABE. Then:
[tex]a=AB=26\\b=AE=6x\\c=BE=2.5x[/tex]
Substituting values and solving for "x", you get:
[tex]26^2=(6x)^2+(2.5x)^2\\\\676=36x^2+6.25x^2\\\\\sqrt{\frac{676}{42.25}}=x\\\\x=4[/tex]
Therefore, the lenghts of the diagonals are:
[tex]AC=12(4)\ cm=48\ cm[/tex]
[tex]BD=5(4)\ cm=20\ cm[/tex]
