Question

If the sum of the diagonals of a rhombus is $12cm$ and its perimeter is $8\sqrt{5}cm$ then the lengths of the diagonals are:

• A. $6cm$ and $6cm$
• B. $7cm$ and $5cm$
• C. $8cm$ and $14cm$
• D. $9cm$ and $3cm$

Answer 1

Answer: (C)

$8cm$ and $14cm$

Perimeter$=8\sqrt{5}$

$4c=8\sqrt{5}\Rightarrow c=2\sqrt{5}$

$2a+2b=12$ (sum of diag.)

$a+b=6\to a$

(Squaring on both sides)

$a^2+b^2+2ab=36\to 1$

Now, $a^2+b^2=c^2$ (Rt angle triangle)

$2ab=36-{\left(2\sqrt{5}\right)}^2$

$ab=8\to b$

From $a$ and $b$

$a=4;b=2$

Length of diagonals are $8cm$ and $14cm$.