Question

Three congruent circle with centre $P,a$ and $R$ are tangent to the sides of rectangle $ABCD$ as show the circle with centre at $Q$ has $Q$ diameter $5cm$ and passes through the points $p$ and $R$. Find the area of the rectangle $ABCD$.

Answer 1

Given:Three congruent circles with centers $P,Q,R$ tangent to circle of rectangle $A,B$ and $O$ circles with center at $Q$ has diameter $=5cm$

Because $PR=5cm$ and $PQ$ is radius of circles centred at $R$

$\therefore SP=PQ=\frac{PR}{2}\Rightarrow \frac{5cm}{2}=2.5cm$

also $RT=QR=\frac{PR}{2}\Rightarrow \frac{5cm}{2}\Rightarrow 2.5cm$

$\therefore ST=2.5cm+2.5cm+5cm=10cm$

and $ST\parallel CD\parallel AB$. [Also, $ST=CD=AB$)

[ Because circles are tangent to the rectangle also centres of them are linear and radius same]

$\therefore AB=CD=ST=10cm$

Since the circles are tangent to rectangle

$AD=$perimeter of circles (identified)

$AD\Rightarrow 2$ (radius) $\Rightarrow 2\left(2.5\right)=5cm$

Area of rectangle $=AB\times AD$

$\Rightarrow 5cm\times 10cm$

$\Rightarrow 50c{m}^2$