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

Answer:

Lenghts of the sides: $26\ cm$

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

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:

$AB=BC=CD=DA=\frac{104\ cm}{4}= 26\ cm$

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

Then, let the diagonal AC be:

$AC=12x$

This means that AE is:

$AE=\frac{12x}{2}=6x$

And let the diagonal BD be:

$BD=5x$

So BE is:

$BE=\frac{5x}{2}=2.5x$

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:

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

Therefore, the lenghts of the diagonals are:

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

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