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

Answer: 5

$C_1{C}_2=r_1-r_2=10$
$r_1=d_1$
$r_2=d_2$
$\Delta =\frac{1}{2}{d}_1{d}_2$ ..... (i)

$\frac{d_1^2}{4}+\frac{d_2^2}{4}=100$
$\Rightarrow d_1^2+d_2^2=\mathrm{400....}(\ast )$
$\therefore d_1-d_2=\mathrm{10....}(\ast \ast )$
From $(\ast )\&(\ast \ast )$, we get
$2d_1{d}_2=300$
$\Rightarrow d_1{d}_2=\mathrm{150....}$(ii)
From (i) \& (ii), we get
$ar(\square ABCD)=\frac{1}{2}\times 150{\text{ unit }}^2$
$=75{\text{ unit }}^2$
$\therefore \frac{ar(\square ABCD)}{15}=5{\text{ unit }}^2$