Question

A rhombus $ABCD$ has sides of length $10$ . A circle with center '$A$' passes through $C$ (the opposite vertex) likewise, a circle with center $B$ passes through $D$. If the two circles are tangent to each other, find the $\frac{1}{15}$ of area of the rhombus.

Answer 1

$C_1 C_2 = r_1 - r_2 = 10$
$r_1 = d_1$
$r_2 = d_2$
$\Delta = \frac{1}{2} d_1d_2$ ..... (i)

$\frac{ d _{1}^{2}}{4}+\frac{ d _{2}^{2}}{4}=100$
$\Rightarrow d_{1}^{2}+d_{2}^{2}=400 .... (*)$
$\therefore d_{1}-d_{2}=10 .... (**)$
From $(*) \&(* *)$, we get
$ 2 d _{1} d _{2}=300$
$\Rightarrow d _{1} d _{2}=150....$(ii)
From (i) \& (ii), we get
$ar(\square ABCD )=\frac{1}{2} \times 150 \text { unit }^{2} $
$=75 \text { unit }^{2} $
$\therefore \frac{ar(\square ABCD )}{15}=5 \text { unit }^{2}$