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: 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]